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QUANTUM TRANSITIONS IN ANTIFERROMAGNETS 
AND LIQUID HELIUM-3 



By 
BRIAN C. WATSON 



A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL 

OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT 

OF THE REQUIREMENTS FOR THE DEGREE OF 

DOCTOR OF PHILOSOPHY 



UNIVERSITY OF FLORIDA 
2000 



ACKNOWLEDGEMENTS 

I would like to thank my thesis advisor. Professor Mark Meisel, for his guidance 
and encouragement over the past three and one-half years. I gratefully acknowledge the 
members of my supervisory committee, Professors Art Hebard, Kevin Ingersent, Yasu 
Takano, and Dan Talham. In addition, I am grateful to Fred Sharifi for sharing his 
knowledge and experience with me. There are several additional people that have 
contributed to this thesis, and I am indepted to Dr. Naoto Masuhara for his enlightening 
physics conversations and Dr. Jian-sheng Xia for his technical acumen. I am also grateful 
to Drs. Stephen Nagler and Garrett Granroth for their assistance with the neutron 
diffraction experiments. Garret Granroth also deserves thanks for teaching me the 
laboratory basics during the semester that we both worked together. Stephen Nagler has 
also contributed to this thesis by writing portions of the MATLAB fitting routines. Every 
member of the Department of Physics Instrument Shop has been extremely helpful. I 
would especially like to thank Bill Malphurs for his attention to detail and for noticing 
when my instrument designs were geometrically impossible. Dr. Valeri Kotov deserves 
thanks for his guidance during my foray into theoretical physics. I am grateful to Larry 
Frederick and Larry Phelps in the Department of Physics Electronics Shop for their 
support. Once again, I acknowledge invaluable input from Professor Dan Talham and the 
members of his research group, including Gail Fanucci, and Jonathan Woodward for 



operating the EPR spectrometer, as well as Melissa Petruska, Renal Backov, and Debbie 
Jensen for synthesis of the antiferromagnetic materials studied in this dissertation. The 
assistance from Dr. Donovan Hall during experiments at the National High Magnetic Field 
Laboratory was invaluable. I would also like to thank Professors Gary Ihas and Dwight 
Adams for loaning equipment and for their help during experiments at Microkelvin 
Laboratory. 



lU 



TABLE OF CONTENTS 

ACKNOWLEDGEMENTS ii 

ABSTRACT vii 

CHAPTERS 

1 INTRODUCTION 1 

1.1 BPCB 2 

1.2 MCCL 4 

1.3 Zero Sound Attenuation in Normal Liquid ^He 5 

1.4 Measurement of the 2 A Pair Breaking Energy in Superfluid ^He-B • • ■ 7 

2 EXPERIMENTAL TECHNIQUES 9 

2.1 SQUID Magnetometer 10 

2.2 Vibrating Sample Magnetometer 12 

2.3 AC Susceptibility 16 

2.4 Tunnel Diode Oscillator 19 

2.5 Conductivity 24 

2.6 Neutron Scattering 26 

2.7 Nuclear Magnetic Resonance 28 

2.8 Electron Spin Resonance 36 

2.9 Pulsed FT Acoustic Spectroscopy 36 

3 THEORETICAL TECHNIQUES 48 

3.1 Exact Diagonalization 49 

3.2 The XXZ Model 57 

4 STRUCTURE AND CHARACTERIZATION OF A NOVEL 

MAGNETIC SPIN LADDER MATERIAL 61 

4.1 The Structure and Synthesis of BPCB 63 

4.2 Low Field Susceptibility Measurements 67 

4.3 Low Field Magnetization Measurements 82 

4.4 High Field Magnetization Measurements 92 

4.5 Universal Scaling 113 

4.6 Neutron Scattering 119 



IV 



5 MAGNETIC STUDY OF A POSSIBLE ALTERNATING 

CHAEsf MATERIAL 127 

5.1 Structure and Synthesis of MCCL 128 

5.2 Electron Paramagnetic Resonance 132 

5.3 Low Field Susceptibility Measurements 139 

5.4 High Field Magnetization Measurements 155 

6 ZERO SOUND ATTENUATION NEAR THE QUANTUM 
LIMIT IN NORMAL LIQUID ^HE CLOSE TO THE SUPERFLUID 
TRANSITION 162 

6.1 Experimental Details 165 

6.2 Zero Sound 177 

6.3 First Sound 187 

6.4 Error Analysis and Final Results 198 

7 DIRECT MEASUREMENT OF THE ENERGY GAP OF 

SUPERFLUID ^HE-B IN THE LOW TEMPERATURE LIMIT 212 

7.1 Details of the FT Spectroscopy Technique 216 

7.2 Thermometry Issues 219 

7.3 Edge Effects 221 

7.4 Temperature Dependence 225 

7.5 Pressure Dependence 23 1 

7.6 Error Analysis 241 

7.7 Absolute Attenuation 242 

8 SUMMARY AND FUTURE DIRECTIONS 245 

8.1 BPCB 245 

8.1.1 Summary 245 

8.1.2 Future Directions 246 

8.2 MCCL 246 

8.2.1 Summary 246 

8.2.2 Future Directions 248 

8.3 Zero Sound Attenuation in ^He 248 

8.3.1 Summary 248 

8.3.2 Future Directions 249 

8.4 Measurement of the 2A Pair Breaking Energy in Superfluid ^He-B • • • 250 

8.4.1 Summary 250 

8.4.2 Future Directions 251 

8.5 Concluding Remarks 252 

APPENDICES 

A LOW TEMPERATURE PROBE DRAWINGS 253 

B COIL FORMER DRAWINGS 271 

C PRESSURE CLAMP DRAWINGS 277 

D NMR PROBE DRAWINGS 286 



E POLYCARBONATE SAMPLE SPACE DRAWINGS 333 

F APPLESCRIPT ROUTINES 338 

G ORIGIN SCRIPTS 344 

H MATLAB FITTING PROGRAMS 347 

I DATA SET PARAMETERS FOR THE LANDAU 

LIMIT EXPERIMENT 350 

LIST OF REFERENCES 354 

BIOGRAPHICAL SKETCH 362 



VI 



Dissertation Presented to the Graduate School 
of the University of Florida in Partial Fulfillment of the 
Requirements for the Degree of Doctor of Philosophy 



QUANTUM TRANSITIONS IN ANTIFERROMAGNETS 
AND LIQUID HELIUM-3 

By 

Brian C. Watson 
December 2000 



Chairman: Mark W. Meisel 
Major Department: Physics 

Effects arising fi-om quantum mechanics are increasingly common in new devices 
and applications. Two different, but related, topics, low-dimensional antiferromagnets and 
liquid ^He, have been studied to obtain a deeper understanding of the quantum mechanical 
properties that govern these systems. Low dimensional magnetism provides a means of 
investigating new quantum phenomena arising from magnetic interactions. Superfluid and 
normal liquid ^He exist in a very pure form and therefore allow severe tests of theoretical 
descriptions. More specifically, the magnetic properties of bis(piperidinium) 
tetrabromocuprate(II), (C5Hi2N)2CuBr4, otherwise known as BPCB, and 
catena(dimethylammonium-bis(n2-chloro)-chlorocuprate), (CH3)2NH2CuCl3, otherwise 
known as MCCL, have been measured and are reported herein. Theoretically predicted 



Vll 



scaling behavior has been observed, for the first time, in BPCB. In superfluid He, the 
pair-breaking edge has been measured at low temperature, thereby allowing for a 
measurement of 2A. These data indicate that the energy gap at low pressure is 
significantly less than predicted by BCS theory. Finally, subtle effects due to the 
attenuation of zero sound in normal liquid ^He have been measured. Evidence for the 
quantum correction to zero sound attenuation, predicted by Landau over 40 years ago, is 
presented herein. 



viu 



CHAPTER 1 
INTRODUCTION 



Quantum mechanical properties of various systems are increasingly important both 
fundamentally and technologically as new materials and devices are being generated at the 
boundary between the classical and quantum worlds. This dissertation addresses the 
quantum mechanical properties of two apparently disparate systems, gapped 
antiferromagnets and ^He. In both cases, however, the quantum mechanical nature of 
these systems is apparent in their macroscopic properties. In fact, these two systems are 
models for the verification of various quantum mechanical predictions. 

The dissertation is arranged as follows. Chapters 2 and 3 detail the nine 
experimental and two main theoretical techniques that were used to collect and analyze the 
data presented herein. Chapter 4 reports the experimental results concerning the gapped 
antiferromagnetic material bis(piperidinium)tetrabromocuprate(II), (C5Hi2N)2CuBr4, 
otherwise referred to as BPCB, and Chapter 5 discusses the alternating chain material 
catena(dimethylammonium-bis(|a2-chloro)-chlorocuprate), (CH3)2NH2CuCl3, otherwise 
referred to as MCCL. These chapters include magnetization and neutron diffraction data 
fi-om experiments at the National High Magnetic Field Laboratory (NHMFL) and Oak 
Ridge National Laboratory (ORNL), respectively. In addition, electron paramagnetic 
resonance (EPR) measurements performed by Professor Talham's research group in the 
Department of Chemistry at the University of Florida are included. The liquid ^He studies 



are presented in Chapters 6 and 7 which discuss the low temperature acoustic experiments 
performed in the University of Florida Microkelvin Laboratory. The first ^He experiment 
is an absolute measurement of zero sound attenuation in ^He above the superfluid 
transition temperature. The second ^He experiment uses Fourier transform techniques and 
is a measurement of the 2A pair breaking energy in superfluid ^He-B. The final chapter 
summarizes the experimental results and lists possible fiiture experiments. 

1.1 BPCB 

The objective of this work is to better understand quantum phase transitions in 
antiferromagnets. Low dimensional, gapped, insulating, antiferromagnetic materials are 
ideal candidate systems for the experimental realization of quantum phase transitions. 
These transitions are defined as phase transitions that occur in the low temperature limit 
{T-* 0), where quantum fluctuations have energies larger than thermal fluctuations 
(ho) > ksT), and are driven by a change in some aspect of the system other than 
temperature. A current review of quantum phase transitions is given by Sondhi et al. [1]. 
When thermal and quantum fluctuations are equally important (ho) ~ kaT), the state of the 
system is referred to as being in a quantum critical regime. Quantum critical behavior is 
important in two dimensional antiferromagnets, and the behavior of charge and spin 
density waves in the quantvim critical regime of two dimensional doped antiferromagnets is 
observed to play a role in high Tc superconductivity [2]. 

To better understand the quantum critical behavior in two dimensional materials, 
we begin by studying quasi-two dimensional systems. The logical intermediate step 
between two dimensional planes and one dimensional chains are ladder materials. The 



long range order that occurs in a two dimensional plane of spins can be approximated by 
ladders of increasing width [3]. Ladders are formed by two or more one dimensional 
chains arranged in a ladder geometry with electronic spins at the vertices of the ladder 
interacting along the rungs of the ladder with exchange Jj. and along the legs of the ladder 
with exchange J\\. In order to further the analogy between the two dimensional cuprate 
high Tc superconductors and quasi-two dimensional ladders, we choose to study ladder 
systems with Cu^^ 5=1/2 spins. Ladders with an even number of legs are expected to 
have a gap to magnetic excitations otherwise referred to as a spin gap, A [3]. A spin gap 
can be measured indirectly in nuclear magnetic resonance experiments or directly in 
neutron scattering experiments. In addition, a spin gap will manifest itself in 
magnetization studies at low temperature (7 ^ 0) as a critical field, Hci, below which the 
magnetization is zero. Recent studies have revealed a connection between the spin gapped 
state in ladder materials and superconductivity [4-6]. 

Until now, the best experimental realization of a 2-leg ladder was thought to be the 
material Cu2(l,4-diazacycloheptane)2Cl4, otherwise known as Cu(Hp)Cl [7-14]. 
However, the low temperature properties of Cu(Hp)Cl have been recently debated [15- 
19]. Although quantum critical behavior has been preliminarily identified in Cu(Hp)Cl 
near Hci, this assertion is based on the use of scaling parameters derived by fitting the data 
rather than the ones predicted theoretically. Clearly, additional physical systems are 
necessary to test theoretical predictions of 2-leg 5=1/2 ladders including quantum critical 
behavior. 

Chapter 4 in this dissertation describes the investigation of the gapped 
antiferromagnetic S = 111 ladder material bis(piperidinium)tetrabromocuprate(II), 



(C5H,2N)2CuBr4, otherwise referred to as BPCB. In 1990, the room temperature crystal 
structure of BPCB was determined, in an x-ray scattering study by Patyal et al. [20], to 
resemble a 2-leg ladder. This crystal structure has been recently verified in neutron 
diffraction experiments. In addition, magnetization and EPR measurements have 
elucidated details of the magnetic exchange. Finally, evidence for quantum critical 
behavior in this material is presented. 

1.2 MCCL 
The simplest antiferromagnetic low dimensional materials are electronic spins 
arranged in one dimensional chains with a single exchange constant between spins, J. An 
exact solution of the isotropic 5=1/2 one dimensional chain was provided by Bethe [21] 
in 1931 for the isotropic nearest neighbor case. In 1983, Haldane [22] predicted a gap in 
the spin excitation spectrum or spin gap for isotropic integer spin chains. The Haldane gap 
for both S = 1 [23,24] and S = 2 [25] systems has been experimentally observed. Spin 
gaps may also occur in half integer spin chains if the exchange between spins alternates 
between two values, J/ and J2, where, to leading order, | J, - ^2 1 ^ ^^b. 

Chapter 5 presents the results concerning the alternating chain material 
catena(dimethylammonium bis(|a2-chloro)-chlorocuprate), (CH3)2NH2CuCl3, otherwise 
referred to as MCCL. The room temperature crystal structure of MCCL was determined 
in 1965 [26] to consist of 5 = 1/2 Cu^^ spins arranged in isolated zig-zag chains with 
adjacent chains separated by (CH3)2NH2 groups. The distance between spins alternates 
between two values and the bond angle between spins is approximately 90 degrees. 
Consequently, the magnetic structure is expected to be an antiferromagnetic alternating 



chain with the exchange constant alternating between the values J\ and Ji. Preliminary 
neutron diffraction work at ORNL has verified the crystal structure. In addition, a 
structural transition has been observed at approximately 250 K and the possibility exists 
for a second structviral transition occurring between 1 1 and 50 K. A description of the 
magnetic exchange is obtained by analyzing the results of magnetization and EPR 
experiments. 

1.3 Zero Sound Attenuation in ^He 
In 1956, Landau advanced a theory based on the properties of normal Fermi 
liquids, and this description is commonly referred to as Fermi Liquid Theory [27,28]. In 
the 1960's, it was realized that ^He at low temperatures was a model system for 
verification of this theory. At this same time, the experimental apparatus needed to study 
^He below 100 mK became available. Since then, this theory has afforded an extremely 
accurate description of the properties of ^He. Landau Fermi Liquid Theory describes a 
perfect Fermi gas, where the interactions between atoms are added as a perturbation. 
These interactions are included by considering elementary excitations with effective mass, 
m*, which are termed quasiparticles. There are two primary modes of sound propagation 
in ^He depending on the time between quasiparticle collisions, r oc M'f, and the angular 
frequency of the sound, ca At high temperatures {on « 1), quasiparticle collisions 
provide the restoring force and the sound propagation is termed hydrodynamic or first 
sound. Consequently, the viscosity and therefore the attenuation of first sound decrease 
roughly with the square of temperature. At low temperatures {cor » 1), quasiparticle 
collisions can no longer provide the necessary restoring force to propagate hydrodynamic 



sound. Instead, sound is transmitted, through quasiparticle interactions, as a collective 
mode by an oscillatory deformation of the Fermi sphere and is referred to as collisionless 
or zero sound. The attenuation in the zero sound regime increases as the square of 
temperature since the relaxation rate of this collective mode increases due to quasiparticle 
collisions. At temperatures well below the Fermi energy {T « Tf), and above the 
superfluid transition temperature, {T> Tc), the attenuation of both first and zero sound are 
well described by Landau Fermi Liquid Theory. 

In the zero sound regime, the attenuation is dominated by scattering within a 
continuous band of quasiparticle energies near the Fermi energy, AE = Ef± ksT. At high 
fi-equencies {ksT « hco « ksTF) collisions will scatter quasiparticles to unoccupied 
energy levels greater than keT away fi-om the Fermi energy. This quantum scattering 
produces a second term in the attenuation, and the attenuation of zero sound may be 
written as 



a,io),T,P)=a'(P)r 



1 + 






(1.1) 



Because the second term is effectively temperature independent, determination of this term 
requires measurement of the absolute attenuation in the zero sound regime. 

Several attempts have been made to verify this second term [29-31], and the most 
recent effort was reported by Granroth et al. [32]. In the latest experiment, the 
temperature and pressure were held fixed while the frequency, /= coIln, was swept fi-om 8 
to 50 MHz. To provide absolute attenuation, the received signals were calibrated against 



the attenuation in the &st sound regime. The result of this measurement was that the 
frequency dependence of the quantum term was a factor of 5.6 + 1.2 greater than the 
prediction. However, the frequency range was limited by the polyvinylindene flouride 
(PVDF) transducers that were used so that/„ax ~ 50 MHz. By extending the experiment 
to higher frequencies, it should be possible to more accurately determine the quantum 
term. However, using the first sound regime as a means of calibration places an important 
restriction on the highest usefiil frequencies. For example, this type of calibration was not 
possible in most other reports [29-31]. 

The objective of this work is to measure the absolute zero sound attenuation in 
^He as a fianction of frequency. In this experiment, relatively low-Q, crystal LiNbOa 
transducers were used to extend the frequency range to approximately yma;c -110 MHz. 
Again, absolute calibration of the attenuation was determined using measurements in the 
first sound regime. For both the zero and first sound data, the temperature was held fixed 
while received signals were averaged at several discrete frequencies. Chapter 6 contains a 
complete description of the results and the analysis at the pressures of 1 and 5 bars. 

1.4 Measurement of the 2A Pair Breaking Energy in Superfluid ^He-B 
The pairing energy of Cooper pairs in the superfluid, 2A, can been estimated in the 
limit of weak coupling using BCS theory [33,34] as 3.5 kBTc, where Tc is the transition 
temperature from the normal to the superconducting state. Deviations from BCS theory 
have been introduced by Serene and Rainer [35] who used quasi-classical techniques to 
incorporate strong coupling corrections in a treatment known as weak coupling plus 
(WCP) theory. One of the first attempts to measure 2A(7) was performed by Adenwalla 



et al. [36] in 1989 who worked at TITc > 0.6 and between 2 and 28 bars. In 1990, 
Movshovich, Kim, and Lee [37] measured the 2A pairing energy over a range of pressures 
(6.0 to 29.6 bars) and temperatures (0.3 < TITc < 0.5). However, in both cases, 
experimental limitations required that the results were either dependent on a particular 
temperature scale or involved extrapolation to zero magnetic field. 

The measurement of the pairing energy in superfluid ^He-B using a novel acoustic 
Fourier transform technique [38,39] is described in Chapter 7. Both the temperature and 
pressure dependences of the 2A pair breaking energy are included. In addition, 
comparisons are made with the existing BCS and WCP plus theory as well as the results 
fi"om previous experiments. 



CHAPTER 2 
EXPERIMENTAL TECHNIQUES 

In this chapter, the experimental techniques employed to study both He and the 
antiferromagnetic materials are discussed. The first three Sections, 2.1 through 2.3, 
describe magnetic susceptibility measurements using a SQUID magnetometer, a vibrating 
sample magnetometer (VSM), and AC mutual inductance techniques. The vibrating 
sample magnetometer research was conducted at the National High Magnetic Field 
Laboratory (NHMFL) in Tallahassee, FL. The next two Sections, 2.4 through 2.5, discuss 
tuimel diode oscillator (TDO) and conductivity measurements, respectively. Section 2.6 
describes neutron diffraction measurements that were carried out at Oak Ridge National 
Lab (ORNL), Oak Ridge, TN. Section 2.7 outlines the design of a nuclear magnetic 
resonance (NMR) probe as well as details of the spectrometer and superconducting 
magnet. Section 2.8 describes electron spin resonance (ESR) measurements which were 
performed by Dr. Talham's research group in the Department of Chemistry at the 
University of Florida. Two ^He acoustic spectroscopy experiments are described, and 
both were conducted in the University of Florida Microkelvin Laboratory. Section 2.9 
highlights only the general experimental approach of both ^He experiments whUe leaving 
the details of each experiment to the relevant chapters. 



10 

2.1 SQUID Magnetometer 

The SQUID magnetometer used in our magnetization experiments (model 
MPMS-5S) was from Quantum Design, Inc., San Diego, CA. The system is composed of 
a computer and two cabinets. The &st cabinet houses the electronics and the second 
cabinet contains the liquid He dewar. The SQUID communicates with the computer over 
the IEEE-488 general purpose interface bus (GPIB), and control of the measurement 
system is accomplished using software provided by Quantum Design. The temperature 
and magnetic field can be varied automatically via the computer software. The 
temperature is controlled by two heaters and the flow of cold He gas, and the useful range 
of operation is from 1.7 K to 300 K with an estimated error of less than 0.5% [40]. The 
temperature can be lowered from 4.5 K to 1 .7 K by applying a vacuum over a small liquid 
He reservoir. For practical purposes, the minimum temperature is 1.8 K and typically 
2.0 K was used to decrease the measurement time. The superconducting magnet provides 
reversible field operation over +/- 5.0 T. 

In this commercial device, the measurement is accomplished using a rf SQUID. A 
SQUID consists of a superconducting ring with a weak link or Josephson junction. The 
electrons in the ring form Cooper pairs, which must be described by a single wave 
function. The phase of the electron wave function on either side of the boundary is 
equivalent. Therefore, the flux through the loop is quantized and must be an integer of the 
flux quantum, hlle . A screening current will increase in the ring to enforce this criteria 
until each integer flux quantum is reached. Similarly, the voltage across the boundary will 
oscillate with a period of one flux quantum. Theoretically, the SQUID can measure 
magnetic flux with a resolution less than 1 flux quantimi. However, practical design 



11 

considerations make it impossible to measure the flux directly; e.g. the SQUID must only 
detect the flux due to the sample and not from the magnet. In a rf SQUID, the 
superconducting ring is shielded from the magnet and connected to the pick-up coils with 
an isolation transformer. A rf signal is applied to an electromagnet so that the flux 
through the ring oscillates. A DC bias is also applied, using a feedback loop, so that the 
voltage across the link remains at the single flux quanta condition. This DC bias is 
proportional to the signal from the pick-up coils and therefore the magnetization from the 
sample. 

The samples are mounted on the end of long stainless steel rods and lowered into 
the sample space. The magnetization of the sample is measured by moving the sample 
through the pick-up coils using a microstepping controller. The pick-up coils have been 
wound so that the voltage in the coils is proportional to the second derivative of the 
magnetization. The computer reads the voltage output as a function of position and 
compares it to a theoretical curve using a linear regression technique. This theoretical 
curve depends slightly on the geometry of the sample. The standard curve assumes a 
cylindrical sample. For all our experiments, 48 position steps were used over a 4.0 cm 
scan length. 

The output of the SQUID is given in units of "emu", which is an abbreviation for 
"electromagnetic units" but it is not really an actual unit. The manner in which "emu" is 
used to output the data has led to some confusion. In cgs units, the "emu" is equivalent to 
cm or erg/G [41,42]. Accordingly, the units of molar volume susceptibility can be 
derived from Curie's Law and may be written in unit form as 



12 



,, 2 2 «^^>"b( ) 3 

Z ^ nN^g //g gaM5^ ^ grg emu ^cm. 

V 3k JV ^ kg(erg/K)TiK)cm' gauss^cm' cm' cm'' 



where n is the number of moles, g is the dimensionless Lande g factor, and A'^ = 
Avagadro's number. The units of magnetization can be obtained from a straightforward 
calculation of total spin as 



M = nN^^B = nN^^si-^^^) = emuG = cm'G. (2.2) 

gauss 



2.2 Vibrating Sample Magnetometer 
High field (0 < // < 30 T) magnetization experiments were performed at the 
National High Magnetic Field Laboratory (NHMFL), Tallahassee. These measurements 
used a 30 T, 33 mm bore resistive magnet and a vibrating sample magnetometer (VSM). 
The general setup of the VSM is shown in Fig. 2.1. Powder and single crystal samples 
(/WW 1 00 mg) were packed into gelcaps and held in place at the end of a fiberglass sample 
rod with Kapton tape. The sample rod screws into the VSM head and is locked in place. 
To position the sample in the field center, the height of the VSM head is adjusted until the 
VSM signal is at a maximunL The VSM uses a pair of counter wound pick-up coils (3500 
turns/each, AWG 50). The sample is vibrated at 82 Hz in the center of the pick-up coils 
to generate a signal. This signal from the VSM is sent through a 19 pin breakout box and 
then to a Lakeshore model 7300 VSM controller. The VSM controller 



13 




r- 






\^ 


— 


\ ki 




1 


ISpin 
bieakoi 


Lit box 


EMU 
monitof 




|4EMU| 
VSM contioHer 





h 


\ 






\ 






vacuuai 
pump 










u 






U 




u 



TT 



Figure 2. 1 : Overview of the VSM setup [43]. 



14 

does not have an IEEE interface and, therefore, a Keithley 2000 multimeter reads the 
EMU monitor on the VSM controller and communicates with the computer over the 
GPIB bus. Absolute signal calibration was not necessary during our measurements, 
because we were able to reach saturation magnetization. In addition, at saturation, we 
were also able to measure and subtract a small linear correction with negative slope that 
corresponds to the diamagnetic contribution from the gelcap as well as the diamagnetism 
from the sample. The VSM has a resolution of 10"^ emu and a maximum signal of ~10 
emu. The largest sample signal was at least an order of magnitude below this limit at 30 
T, so the VSM pick-up coil response remained in the linear regime. The sample signal 
was greater than the minimum signal resolution of 10"^ emu at a magnetic field of 
approximately 1 T. 

Temperature control was achieved by varying the pumping speed on either a '^He 
or He bath. A heater was not used in our experiments. The resistance values of a 
calibrated cemox thermometer were measured and converted to temperature using a 
Conductus LTC-20 Temperature Controller. The cemox resistor is calibrated only down 
to a temperature of 2 K. In addition, the cemox resistor has a field dependence that must 
be corrected using the results of Brandt et al. [44]. This thermometer is placed in a 
location directly adjoining the sample space (see Fig. 2.2). The sample space as well as 
the middle layer surrounding the sample space is filled with a small amount of '*He gas. 
When the temperature of the ^He bath falls below the ''He lambda transition temperature, 
some of the gas in the surrounding sample space becomes superfluid and the thermal 
connection with the bath is made. For this reason, above a temperature of 2.0 K, where 



15 



'He bath 

4 

He 
superfluid 



77 K 




T 



150 cm 



cemox 
thermometer 



t 



0.64 cm 



Figure 2.2: A sketch of the thermometer setup and thermal conduction mechanisms below 
the He lambda transition temperature in the NHMFL vibrating sample magnetometer. 



the thermometer is calibrated, the thermometer values were used directly. Below this 
value, the temperature is estimated from the ^He pressure. 

In our experiment, the lowest ^He bath temperature was 0.58 K. The actual 
sample temperature was warmer than this temperature. The main mechanism for thermal 
conduction inside the sample space was a small amount of ''He gas. At such a low 
temperature, an error in the temperature of 0.1 K becomes very important. By estimating 
the heat leak, we can determine the worst possible error in temperature from 



16 



Q^=Qc,zrA (2.3) 



Area k = Area k, (2.4) 

AL radius 



where Qf^ and ^^ ^^ the rate of heat transfer into and out of the sample space and k is 
the thermal conductivity of the ""He gas. The area and radius characterize the inside of the 

0.64 cm diameter tube. The factor — is determined by considering the dewar geometry. 

AI 

The liquid N2 bath temperature of 77 K is 150 cm above the sample. If we assume that 

the temperature gradient is a constant of 0.5 K/cm, then we arrive at a Ar of 0.2 K. This 

calculation is obviously an overestimate; however it gives us a basis for determining the 

maximum possible error. Consequently, the lowest temperature in our experiment, 

originally reported as 0.58 K, was estimated to be 0.7 ±0.1 K. 

2.3 AC Susceptibility 
The AC susceptibility measurement system is a standard mutual inductance 
technique that consists of a dewar, probe, and electronics. Detailed drawings of the probe 
are listed in Appendix A. Computer control of the instruments was accomplished using 
Labview and a GPIB interface. Five instruments were employed: a Picowatt AVS-47 
Resistance Bridge, HP3457a digital multimeter, HP6632 power supply, and two PAR 
124 A lock-in amplifiers. A schematic of the susceptibility setup is shown in Fig. 2.3. The 
signal fi-om the secondary coil is split into inputs A and B of both lock-in amplifiers which 



17 



PAR124ALock-m 

<D + 90° 

Signal 
Ref Out A B Out 



^^ 



PAR 1 24A Lock-in 

Signal 
Refin A B 0»* 



-^3- 



HP3457a 



InputO Inpat 1 



A 



4.5 kn 
Resistor box 



B 




Voltage 



AVS-47 



Current 



Lakeshore carbon 
glass resistor 
CGR-1-1000 



Heater 



Figure 2.3: (A) Schematic diagram of the mutual inductance circuit used to measure AC 
susceptibility. (B) Overview of the copper sample plate indicating the position of the 
thermometer, heater, and susceptibility coil. 



18 

are operated in 'A-B' mode. The inputs are filtered so that high fi-equency components 
if >\ kHz) are attenuated. At the start of the measurement, usually at the lowest 
temperature, the lock-in amplifiers are adjusted so that the signal fi-om one is a maximum 
and the phase difference between them is » 90 degrees. The adjustable reference output 
provides the primary excitation voltage (typically Vref = 5 Vp.p). Because the resistance 
of the primary coil is small (50 Q), a 4.5 kQ resistance box is placed in series with the 
primary so that the reference output behaves as a constant current source. 

The heater consisted of approximately 100 turns of manganin wire on a copper 
core, which was bolted to the copper sample support (Fig. 2.3 B). The resistance of the 
heater was approximately 50 Q. Five watts of heater power was sufficient to bring the 
temperature fi-om 4.2 K to 77 K. The Labview software controlled the heater to achieve 
an input drift rate, typically 0.2 K/minute. The temperature could be lowered fi-om 4.2 K 
down to 1.7 K using the IK pot. Temperature control in this range was achieved either by 
controlling the IK pot pressure manually using the pumping valve or by opening the valve 
all the way and letting the computer control the temperature using the heater. 

The temperature was determined by measuring the resistance of a CGR-1-1000 
Lakeshore Cryotronics carbon glass resistor and converting to temperature. The resistor 
was wrapped in copper shim stock and bolted to the sample plate. The resistor wires were 
attached to the cold plate using GE varnish. The resistance was read using the AVS-47 
resistance bridge on the 2 kQ range using a 1 mV excitation at 15 Hz. 

The susceptibility coil consists of two coil formers, a primary (insert) and a 
secondary (outer). The coil formers were manufactured fi-om phenolic rod and the 
drawings are shown in Appendix B. The primary has 300 turns on two layers using 



19 

copper AWG 40 wire. The secondary consists of two counter wound coils separated by a 
small gap with 1500 turns/side on 40 layers using the same wire. The insert is a tight fit 
into the secondary and the position of the insert is adjusted so that the signal fi-om the 
secondary coil is a minimum. The sample is placed on one side of the insert so that any 
magnetic signal from the sample will unbalance the coil. The susceptibility coil was 
attached to a 16 pin connector using GE varnish and plugged into the sample plate. The 
sample plate is suspended below the IK pot using two stainless steel tubes. 

2.4 Tunnel Diode Oscillator 

The tunnel diode oscillator (TDO) technique uses a tank circuit that is sensitive to 
the inductance of a coil where the sample is located. The sample is placed inside the coil. 
As the inductance of the sample changes, the resonant frequency of the circuit also 
changes. In the case of a conducting sample, this inductance change is related to the skin 
depth. The TDO technique is particularly sensitive to superconducting transitions. For a 
discussion of the superconducting applications, please see the dissertation of Philippe 
Signore [45]. 

The circuit used for the TDO experiment is shown in Fig. 2.4. The tunnel diode 
(Germanium Power Devices, model number BD-6) is the active element in the inductance 
circuit. This device has a negative I-V curve when biased at the correct voltage and 
therefore behaves analogously to a negative resistor. A 5 pF capacitor is added in parallel 
with the tunnel diode to stabilize the oscillation. In series with the tunnel diode are the 
wire capacitance and the coil inductance. The wire capacitance is on the order of 300 pF. 



20 



HP 5385A 
Frequency 
Counter 




Figure 2.4: Schematic diagram of the tunnel diode oscillator setup. 



21 

Additional capacitance can be added to change the resonant frequency which is typically 
10 MHz. 

The coil is wound on a phenolic former fabricated in the University of Florida 
Instrument Shop. The drawings for various designs are listed in Appendix B. For 100 
turns of AWG 40 copper wire, we expect a coil inductance of ~5 ^H. Power is supplied 
by two 1.5 V Mercury batteries at room temperature. The voltage bias is adjusted until 
the circuit begins to oscillate. It is useful to watch the signal output on an oscilloscope as 
the voltage bias is adjusted. The voltage where the signal appears most sinusoidal usually 
corresponds to the maximum voltage output. However, because this point usually occurs 
near the end of the voltage range for stable oscillation, it is not practical to balance the 
circuit at this point. It is more useful to balance near the center of the voltage bias curve 
so that changes in the sample inductance, due to a change in temperature, do not push the 
circuit out of its oscillation condition. After the DC component is removed, the signal 
voltage (-50 mVp.p) is increased (~1 Vp.p) by a Trontech W500K rf amplifier. A 
HP5385A frequency counter reads the frequency and communicates with the computer 
using GPIB bus. Temperature control is identical to the method discussed for the AC 
susceptibility technique (Section 2.3). 

A major complication of this method is that the properties of the tunnel diode 
circuit are temperature dependent. Even without a sample, the resonant frequency of the 
circuit will change due to a change in the wire resistance, thermal contraction of the 
sample coil, and most importantly the thermal dependence of the timnel diode. From 
4.2 K to 77 K, this change is typically on the order of a few MHz. Frequency transitions 
related to the sample are at least an order of magnitude less, and a temperature dependent 



22 

background must be subtracted. One possible way to decrease the background 
contribution is to mount only the tunnel diode on the underside of the IK pot. The leads 
must be attached to the copper using GE varnish. As long as the IK pot needle valve is 
open, the temperature of the IK pot will remain stable and no background will need to be 
subtracted. However, at temperatures above 20 K, this method will boil-oflf a 
considerable amount of liquid He from the bath. To avoid this limitation, the wires 
connecting the tunnel diode to the circuit should have low thermal conductivity, such as a 
CuNi alloy. The increased resistance of the wires will lower the output signal slightly. 

It has been observed that the transition temperature of high Tc superconductors 
[46] and hole doped ladder materials [47] can be increased by the application of pressure. 
Two pressure cells have been constructed to study the effects of pressure on the 
superconducting transition temperature using the previously described tunnel diode 
technique. Both pressure clamps are based on the design of J.D. Thompson [48], which is 
an improvement over a previous design [49]. The second design is approximately twice as 
large as the first. The clamp bodies and sample cells are made from beryllium copper and 
Teflon, respectively, at the University of Florida Instrument Shop. The tungsten pushrods 
and wafers are made of tungsten carbide by Carbide Specialties, Waltham, MA. The 
larger cell is shown in Fig. 2.5. Complete drawings for this cell are given in Appendix C. 

The sample coil (~ 1 mm diameter for the smaller cell, 100 turns, 4 layers of AWG 
40 wire) is contained inside the Teflon sample cell. A second coil of the same size is 
usually included and contains lead wire for the pressure calibration since Tc{P) for 



23 



U 



□n 




Ul 





o 



O o 



o 





o <= 



Figure 2.5: Overall drawing of beryllium copper pressure clamp. 



lead is well known [50]. The wires exit through the top of the Teflon cell and are secured 
to the brass lid with 2850 epoxy. The cell is filled with isopentane liquid to distribute the 
pressure. A retaining ring is placed on the bottom of the Teflon cell to prevent the cell 
fi-om rupturing outward. The entire pressure clamp is placed in a press, and the tungsten 
carbide pushrods on either side of the cell transmit the pressure. As the pushrods are 
compressed, the top of the beryllium copper clamp (the side without the wires) is 
tightened. By using the surface area of the Teflon cell and the pressure applied it is 
possible to calculate the actual cell pressure. 



24 
2.5 Conductivity 

As part of a collaboration with Dr. Talham's research group (Department of 
Chemistry, University of Florida), several attempts were made to measure the conductivity 
of Langmuir Blodgett (LB) films deposited onto glass slides. These films are typically a 
few hundred angstroms thick and are expected to be semiconducting. Since the resistivity 
of these fitois was expected to be in the 10 MQ-cm range, masks were engineered to 
optimize the conductivity measurements. The size and dimensions of the masks used are 
shown in Fig. 2.6. The masks were cut fi-om stainless steel sheets using an electric 
discharge machine (EDM) in the Department of Physics Instrument Shop. 

The transport measurements were performed in the following manner. First, a 
mask was placed onto the LB film, the bottom of the glass slide was glued to the metal 
"puck" using rubber cement, and the whole assembly was placed in the evaporation 
chamber. During the evaporation, the sample is inverted. To prevent the mask fi-om 
falling, the mask is held in place by gluing the sides of the mask (which hang over the 
sample) to additional glass slide pieces using rubber cement. The evaporation chamber 
was placed under vacuum using a combination mechanical/difiusion pump system. When 
the pressure reached 1x10"^ Torr (after approximately three hours), the evaporation 
would begin. The current source was set to 180 A, giving a gold evaporation rate of 
2 A/s. Typically, 300 A of gold were deposited. After removal from the evaporation 
chamber, the glass slide was attached to a GIO support using rubber cement. Using silver 
paint, gold wires were attached as current and voltage leads to the evaporated gold. 
These wires were also glued to flat copper contacts on each edge for strain rehef The 
circuit diagram for the four-probe technique is shown in Fig. 2.7. Measurements were 
done at 19 Hz and used an initial excitation voltage of 200 ^iV. If the resuh was infinite 
resistance, the excitation voltage was increased to 20 mV. In spite of considerable effort, 
no reasonable resistance ( < 100 MQ) was obtained. The samples were expected to be 



25 



0.010 cm 




(A) 



(B) 



Figure 2.6: Stainless steel mask designs used for gold evaporation onto Langmuir 
Blodgett films. 



semiconducting, so warming the samples should have increased the carrier density and 
hence the conductivity. Each sample was placed under a lamp to increase the temperature 
up to 100 C above ambient. Efiforts were also made to improve gold contact with the 
conducting layer of the LB sample through gold "scarring". To rule out the possibility 
that the contact resistance was unusually large, the contact resistance was measured using 
the mask in Fig. 2.6 (A). In every case, the contact resistance was on the order of 0.5 Q, 
and the sample resistance was infinite. 



26 



G10 Support copper shim stock 



super 
glue 



LB Film 
glass slide 




LR-700 
Resistance Bridge 




(A) 



(B) 



Figure 2.7: Circuit diagram for four-probe AC measurements on LB film samples. The 
figure in (A) is a magnified view of the LB film while (B) is an overview of the entire 
experimental setup. 



2.6 Neutron Scattering 
Neutron scattering experiments were performed at the Oak Ridge National 
Laboratory High Flux Isotope Reactor (HFIR). When the reactor is operating at fiill 
power (85 MW), it produces a large thermal neutron flux of 1.5 x lO'^ (neutrons/cm^ sec). 
This flux is accessed by four 10 cm diameter beam tubes that extend horizontally from the 
midpoint of the reactor core. The neutron flux passes through a sapphire filter to limit the 
amount of fast neutrons. At each of the access points, there is a triple axis spectrometer, 
labeled HB-1 through HB-4. All of the experiments listed in this dissertation utilized the 



27 

HB-3 spectrometer (see Fig. 2.8). The typical monochromatic neutron flux (resolution 
~ 1 meV) after collimation is 3 x 10^ (neutrons/cm^ sec) [51]. The angle between the 
sample and the incident and reflected beams can be changed independently. Changing the 
20M monochromator angle can continuously vary the incident energy. After interacting 
with the sample, the neutron beam is defi^acted by the analyzer which is usually the same 
material as the monochromator. It is also possible to operate without an analyzer, which 
essentially allows all final neutron energies. After the analyzer, a ^He detector registers 
the neutron flux. 

Both q scans at integrated final energy and AE scans at fixed q were performed. 

For both types of scans, the monochromator was pyrolitic graphite (PG)[002] with a fixed 
incident energy of 14.7 meV or 30.5 meV. The collimation was 20' at positions C2 and 
C3, before and after the sample. A collimation of 60' represents the open beam which has 
dimensions of 5 cm x 3.75 cm. A PG[002] filter was also used to remove second order 
reflections. For the constant AE scans at fixed q , a PG[002] analyzer was used to select a 
fixed final energy. With the collimation, the resolution was typically 1 meV. The BPCB 
sample was a single deuterated crystal with the approximate dimensions of 10 mm x 7 mm 
X 2 mm, while the MCCL sample was approximately 1 g of deuterated powder. The 
specimen was attached to a sample support which also served as a thermal anchor. 
Temperature control was accomplished by varying the pumping speed on an in-house '*He 
cryogenic system. 



28 



HB-3 Spectrometer shutter 



Collimator (CI) 



Collimator (C2) 




Sapphire RIter 
HB-3A Beam 



Sample 



Monochromator 



Analyzer Crystal 
Collimator (C4) 

^He Detector 



Figure 2.8: Overhead view of HB-3 beamline at the HFIR facility at ORNL [51]. 



2.7 Nuclear Magnetic Resonance 
The nuclear magnetic resonance system can be divided into four main sections: 
finger dewar, probe, spectrometer, and superconducting magnet. The drawings for the 
dewar, probe, and magnet are included in Appendix D. The finger dewar was purchased 
fi-om Kadel Engineering, Danville, IN, and was designed specifically for our Oxford 
magnet. The overall length of the body is 49 inches and the tail portion is 33 inches long. 
It contains a liquid N2 shield with a 5.7 liter capacity designed to have a 2.5 day hold time 
and a liquid He capacity of 35 liters with a 13.2 day hold time. A stainless steel O-ring 



29 

flange bolts to the top and necks down to a LF flange (A&N, part number 
LF100-400-SB). Sbc copper baffles are suspended below this flange along with three 4-40 
stainless steel support rods. There are seven quick connects (A&N, part number QF16- 
075) arranged in circle 45 degrees apart for access to the He bath. 

The NMR probe was designed to be a versatile platform to study protons as well 
as other nuclei. It is based loosely on a design by A.P. Reyes and co-workers [52]. The 
main additional objective of this design was to allow the ability to tune the NMR 
capacitors from room temperature while the probe vacuum can was cold. In addition, the 
probe should allow for stable temperature control between 1.5 K and 300 K. The 
combination of these goals presents another difficulty, i.e. ^He gas exists as single atoms 
and therefore does not contain any molecular vibration modes. Therefore, He has a low 
ionization energy with a minimum at ~1 x 10"^ Torr [53], which is incidentally the typical 
vacuum pressure produced by a mechanical pump. At high power (P > 100 W), current 
will arc between the terminals of each capacitor. To overcome this difficulty, the He 
vacuum pressure must be higher than 1 Torr or lower than 1x10"^ Torr. For the IK pot 
to be effective, the lower pressure must be chosen. Although this pressure is easily 
reached using a turbo or difiusion pump, the lack of any exchange gas means that the 
sample must be thermally anchored by other means. We have chosen two ways to solve 
the capacitor arcing dilemma. First, the sample can be contained in a separate 
polycarbonate sample space (see Appendix E). Second, the capacitors can be contained in 
their own vacuum cans and pumped independently. The flexibility in the design allows us 
to switch between the two methods or use a combination of them. In the following 
paragraphs, I will discuss the notable features of the NMR probe design. 



30 

Machining of the probe parts as well as the probe assembly took place at the 
Department of Physics Instrument Shop. The top of the probe is built on a LF flange that 
mates to the stainless steel adapter on the top of the dewar (Fig. 2.9). The electronic 
connections are made through the brass box at the top of the probe. The main pumping 
line is also connected to this box with a quick connect flange. There are five stainless steel 
rods that connect the room temperature flange to the vacuum can: the main pumping Une, 
two capacitor pumping lines, the IK pot pumping line, and the needle valve control rod. 
The horizontal position of the IK pot pumping line is shifted near the top of the baffles, 
using a brass adapter, to prevent a direct line of sight fi-om room temperature. The IK pot 
pumping line also contains its own set of internal baffles. At the junction of the pumping 
line and the IK pot, the design has been carefully chosen to prevent superfluid ''He from 
climbing the pumping line walls. The IK pot is located near the bottom of the vacuum can 
and filled from a 0.050 inch OD capillary connected to the needle valve. 

The capacitor pumping lines also accommodate the capacitor adjustment rods. 
These capacitor control lines are fabricated from a combination of 1/8 inch diameter 
stainless steel and phenolic rod. Each rod is fitted with triangular phenolic spacers so that 
it remains in the center of its pumping line and ends with a stainless steel screwdriver tip 
designed to fit the variable capacitor. The phenolic portion of the rod is located at the 
bottom to prevent any increase in capacitance. Each control rod must be spring loaded 
because the height of the capacitor changes upon rotation. A double 0-ring seal at the top 
of each rod allows adjustment of the capacitors without significant increase in vacuum 
pressure at 1 x 10* Torr. 



31 



3/8 CajuF 




Needle Valve 
Control Rod 



Main Pumping Line 




Baffle 



Double O-ring 
Seal 



LF Flange 
1K Pot Pumping Line 



1KPot 



Sample 
Suppo 



-Capacitor Adjustment 
Rods SampI 

Space 

Baffle 




1 .69 inches 



Capacitor 
Can 



(A) (B) 

Figure 2.9: Detail of the NMR probe top (A) and bottom (B). 



The capacitors are 40 pF non-magnetic trimmer capacitors from Voltronics, 
Denville, NJ (part number NMTM38GEK). The position of the capacitors has been 
chosen so that the distance from the NMR sample coil is minimized. The frequency range, 
over which the NMR circuit can be tuned, is limited by this length. Each capacitor is 
screwed into the bottom of its pumping line using a Vespel spacer to prevent any electrical 
connection with the stainless steel can. As mentioned earlier, each capacitor can be sealed 
in its own individual vacuum can using soft solder. 



32 

The vacuum can is 28 inches long so that the vacuum can top flange is higher than 
the dewar neck. The vacuum seal is achieved using indium wire. The distance between 
the vacuum can outer wall and the dewar inner wall is only 0.055 inches. During a magnet 
quench or some other unforeseen event, the He below the probe would not be able to 
escape easily. A hard Styrofoam piece must be bolted to the vacuum can bottom to 
exclude any He liquid. The vacuum can contains its own set of six copper baffles to limit 
the radiation heat leak from the pumping lines. The CuNi twisted pair for the 
thermometer and heater is thermally connected to the top of the vacuum can and the IK 
pot. The NMR signal is transmitted using semi-rigid coax which is thermally connected at 
those same points using hermetic feedthroughs (Johnson, part number 142-000-003). The 
use of hermetic feedthroughs insures that the inner conductor is also thermally anchored. 
Care has been taken at these points to insure that thermal contraction of the semi-rigid 
coax is permitted. The polycarbonate sample space has a separate 0.025 inch outer 
diameter capillary that runs to the top of the probe. The gas in this capillary is also 
thermally anchored using copper bobbins at the top of the vacuum can and the IK pot. 

The spectrometer is a commercial instrument designed by Tecmag Inc., Houston, 
TX. The spectrometer can be divided into five main components: NMRKIT 11, Libra, 
PTS 500, American Microwave Technology (AMT) Power Amplifier, and G4 PowerMac. 
A continuous wave rf signal is sent from the PTS-500 frequency generator to the 
NMRKIT. The NMRKIT mixes the signal with an intermediate frequency of 10.7 MHz 
(heterodyne detection) and uses only the lower side band. The signal is amplified by the 
AMT Power Amplifier for a maximum pulse power of 400 W. It is important to note that 
the AMT is limited to a maximum frequency of 300 MHz. The in-phase and quadrature 



33 

components are recorded by the Libra with a time resolution of 100 ns. The software 
allows the returned signal to be viewed in either time or frequency space and to perform 
data manipulation. The phase of the transmitter pulse is cycled to decrease the signal due 
to coherent noise. Due to the phase cycling, all sequences must contain a multiple of four 
pulses. The computer communicates with the Libra using a set of control lines cormected 
through a ribbon cable to an internal Tecmag card. The computer is connected to the 
NMRKIT II using a Mini-Din serial cable. Since the G4 PowerMac does not have a serial 
port (the spectrometer was originally designed for an older PowerMac version), a 
Keyspan (Richmond, CA) adapter card was purchased to meet that need. Tecmag has 
provided software that allows pulse sequences to be created. In addition, Applescripts 
[54] can be written to access the spectrometer software and perform automated data 
acquisition. 

A schematic of the pulsed NMR setup is shown in Fig. 2.10. The crossed diodes 
(Hewlett Packard Shottky, part number 1N734) have been chosen with a 2 ns response 
time to insure proper operation up to 300 MHz. The purpose of the crossed diodes in 
series after the power amplifier are to prevent unwanted noise before or after the pulse. 
Use of a directional coupler instead of a "magic tee" is preferred for pulsed applications. 
A directional coupler concentrates the losses to the input side, which can be overcome 
with additional power, while a magic tee divides the losses between the input and output 
[55]. The NMR circuit is balanced to 50 Q prior to connection with the directional 
coupler using a HP8712 Network Analyzer. The cable length between the directional 



34 



PTS500 



I 
s 



Ribbon cable 



Main output 



Serial 
line 



Synth in 
TECMAG 
NMRKIT 



Rfout 



Various 
interconnects 



TECMAG 
Libra 



LJ 



Probe in 



Miteq 

AU-1I14 

pre-amp 



A./4 




Crossed diodes 
in parallel 





|_J Rfout 



Shottky IN734 
crossed diodes 
in series 



GPIB 



TS-530A 



AVS-47 



Twisted 
'pair ^^ 



Mini-Circuits 

15542 

Directional Coupler 



Semi-rigid 
SMA 



Dewar 



balanced to 50 ohms 
using the network analyzer 
(includes cables) 



I Kohm Lakeshore 

I/4W CX-1030-SD 

Metal film Cemox 

chip resistor resistor 




Figure 2. 10: Schematic diagram of the pulsed NMR setup. 



35 

coupler and the pre-amp should be XI A (speed of signal « 2c/3, where c = speed of light). 
The impedance of a XI A transmission line obeys the following relationship: 

2,NP*^om=^^^- (2-5) 

When the pulse reaches the directional coupler, Zqut = 0, due to the crossed 
diodes. The effective input impedance will be infinite along the pre-amp side and 50 Q 
along the NMR circuit side. Consequently, all of the power wiU be directed toward the 
NMR circuit. After return fi-om the NMR coil, Zqut = 50 Q, and the pulse will see a 50 Q 
pre-amp circuit impedance. 

Temperature control is achieved using the Picowatt system AVS-47 resistance 
bridge and TS-530A temperature controller. The G4 PowerMac controls these 
instruments using the GPIB bus and Labview software. Thermometer resistance set points 
are sent to the temperature controller, which adjusts the heater power to achieve those set 
points according to the time constants set by the software. The temperature controller is 
limited to 1 W of heater power which is sufficient for most applications. The thermometer 
is a Lakeshore CX-1030-SD Cemox resistor calibrated from 1.4 K to 100 K. The heater 
is a 1 kQ, 1/4 W, metal fihn chip resistor. Both the heater and thermometer are attached 
to the polycarbonate sample space with Emerson & Cuming 2850 epoxy. 

The Oxford superconducting magnet has an 88 mm bore diameter and a maximum 
center field of 9 T located 570 mm below the top flange. When at the maximum field of 
9 T at the center of the solenoid, the field at the top flange is -0.06 T. There are four 
shim coils to improve the field homogeneity. This magnetic field variation is less than 



36 

6 ppm in a 1 mm diameter spherical volume at the maximum field. The main magnet and 
the shim coils are energized using Oxford power supplies. The power supplies 
communicate with the G4 PowerMac computer over the GPIB bus. It is important to 
shunt or "dump" the shim coils when charging the magnet to full field. Failure to perform 
this step will result in current being trapped in the shim coils, and this excess field may 
cause a quench. 

2.8 Electron Spin Resonance 
The electron spin resonance (ESR) measurements were carried out by Dr. 
Talham's research group at the Department of Chemistry at the University of Florida. The 
ESR spectrometer is a commercial Brueker X-band (9 GHz) spectrometer. Temperature 
control down to 4 K was achieved with an Oxford ESR 900 Flow Cryostat. Further 
details are available by reviewing the dissertation of Garrett Granroth [56]. The ESR 
measurements discussed in this dissertation were made on both powder and single crystal 
samples. Typical ESR spectra consisted of a single broad line v^th a width of 
approximately 500 G. Exact calibration of the Lande g factor for each material was made 
by comparison with the center fi-equency of the fi-ee radical standard DPPH. 

2.9 Pulsed FT Acoustic Spectroscopy 

Pulsed Fourier Transform acoustic spectroscopy experiments on liquid ^He were 

conducted at the Microkelvin Laboratory at the University of Florida. The sample cell 

was placed in a Ag tower mounted on a Cu plate attached to the top of the Cu 

demagnetization stage of Cryostat No. 2. Further details of this cryostat design are 



37 

described by Xu et al. [57]. A Ag powder heat exchanger in the Ag tower provides 
thermal contact for cooling the liquid sample. Miniature coaxial cables with a 
superconducting core and a CuNi braid were used between the tower and the IK pot. 
Stainless steel semi-rigid coaxial cables were used from the IK pot to the room 
temperature connectors. Above 40 mK, the temperature was measured using a calibrated 
ruthenium oxide (Ru02, Dale RC-550) resistor which has approximately a 500 Q room 
temperature resistance [58]. The value of this Ru02 thermometer was measured using a 
Pico watt AC Resistance Bridge (Linear Research). A heater mounted on the nuclear 
stage supplied up to 1 fxW of thermal power. From 40 mK to 1 mK, the temperature was 
measured using a ^He mehing curve thermometer. Below nominally 3 mK, a Pt NMR 
thermometer (PLM-3, Instruments of Technology, Finland) was used and calibrated 
against the ^He melting curve [59]. The pressure was determined using a strain gauge 
mounted next to the tower. Details of the Tecmag commercial spectrometer were 
discussed in the previous section. 

A cross sectional view of the sample cell, nominally a cylinder with a radius of R = 
0.3175 + 0.0010 cm, is shown in Fig. (2.11). In order to obtain a large frequency 
bandwidth, low-Q, coaxial LiNbOa transducers were used and were separated by a 3.22 + 
0.01 mm MACOR spacer and held in place with BeCu springs. The spacing was measured 
before the experiment and verified in situ by measuring the time delay between successive 
zero sound reflected pulses. The transducers had a fimdamental resonant frequency of 
21 MHz and were operated in four frequency windows: 8-12, 16-25, 60-70, and 
105-111 MHz. Figure 2.12 shows the power reflection coeflScient vs. frequency for the 



38 



Upper Signal 
Cable 

CuNi Shield 



Lower Crystal 



|[r^ /Ti 




Ag Cell Body 
BeCu Spring 




Upper Crystal 
MACOR spacer 
CuNi Shield 



Lower Signal 
Cable 



Figure 2. 1 1 : Cross section of the ^He acoustic cell. 



c 
o 

it 

o 
O 

c 
o 

0} 

a: 
o 

Q. 




Receiver 
Transmitter 



Sensitive Range 
of Receiver 



104 106 108 110 112 

Frequency (MHz) 



114 



116 



Figure 2.12: Frequency response functions of the 5"" harmonic for two LiNbOs 
transducers at 0.3 K [39]. 



39 

5' harmonic of each LiNbOa transducer at 0.3 K [39]. The overlap of the transmitting 
and receiving transducer bandwidths determines the useful frequency range at each 
harmonic window. For the case shown in Fig. 2.12, this operational range is 
approximately 3 MHz centered at 108 MHz. Similar results were obtained for the other 
frequency windows. Due to the highly structured resonance peaks in the frequency 
response fimction of each transducer, the resulting signal is structured. 

A schematic circuit diagram of the pulsed FT acoustic technique is shown in 
Fig. 2.13. The strain gauge and the ^He melting curve thermometer (MCT) were both 
monitored using a capacitance bridge (General Radio Company, type 1615-A) and the 
output was sent to a PAR 124A lock-in. The balance of each lock-in was determined by 
measuring the fiinction output voltage using a HP34401A multimeter. The multimeters 
communicated with the G4 PowerMac over the GPIB bus. The pulse output of the 
Tecmag spectrometer was sent to an inline attenuator before reaching the connection 
points on the dewar. Although the pulse output of the spectrometer can be adjusted, it is 
easier to consistently set the voltage level of the input pulse using an attenuator. Because 
the output level of the NMRKIT II was controlled manually by a knob, for repeatability it 
was set to the maximum of 13 dBm at all times. The attenuator was set to either -10 or 
-20 dB. The received signals from the sample cell were amplified approximately 20 dB by 
a Miteq AU-1114 preamp. The Pt NMR thermometer signal was monitored using a 
TDS-430A digital oscilloscope. Each Pt NMR signal trace was sent to the PowerMac 
over the GPIB bus and recorded for later analysis. Labview software was used to 
communicate with the GPIB instruments. However, the spectrometer was operated using 



40 



PAR 124A 

Lock-in Function 
A out 



GRC 

Capacitance 

Bridge 



PAR 124A 
Lock-in 



Funaior 
out 



GRC 

Capacitance 

Bridge 




Figure 2.13: Schematic diagram of the pulse FT acoustic ^He spectroscopy technique. 



41 

commercial software provided by Tecmag. The synchronization between Labview and the 
Tecmag software was accomplished using Applescript routines (Appendix F). Typically 
the spectrometer data at each frequency averaged the results of 128 pulses with a 4 s wait 
step between each pulse. Each transmitter pulse was 0.4 [is (Fourier transform 
spectroscopy) or 4 |as (amplitude/time of flight acoustic spectroscopy), depending on the 
experiment, and the pulse power at the sample cell was estimated to be approximately 
-20 dBm. The waiting time between different frequencies was at least 8 minutes. The 
real and imaginary components were separately digitized at 10 M samples/s for a 2048 
samples. Spurious signals, resulting from electrical crosstalk, appeared in the first 
microsecond of data. Before taking the FT, this region of the data was blanked (set to 
zero). In addition, for some experiments, echoes were eliminated by blanking to avoid 
adding spurious structure in the frequency spectrum. This blanking and subsequent FT 
were accomplished using Origin scripts (Appendix G). Calibration of the MCT involved 
several steps. In the first step, the ^He MCT pressure was changed using a standard 
zeolite absorption ^He pressure bomb (i.e. "dipstick") and the resulting pressure was 
measured using a Digiquartz transducer. Figure 2.14 shows the result of a capacitance vs. 
pressure calibration of the melting curve thermometer using this method. This step should 
be accomplished at a relatively warm temperature (-150 mK) to decrease the amount of 
time needed to reach equilibrium when changing pressures. The calibration of the strain 
gauge was accomplished using a similar technique. Once the capacitance vs. pressure 
calibration is complete, a predetermined temperature vs. pressure curve is required. The 
experiments in this dissertation used the ^He melting curve of Wenhai Ni (Fig. 2.15 [59]), 
which is consistent v^dth the Greywall scale [60]. 



42 



32 


_ o Experimental Points ^ 




1 oiynomiai rii c^ 


g 31 


y^ 


Capacitance 

M CO 
CO O 


'- X 


28 


1(1.1.1,1,1, 



28 29 30 31 32 33 

Pressure (bars) 



34 



35 



Figure 2.14: The capacitance vs. pressure of the MCT at 150 mK. The solid line is a fifth 
order polynomial fit to the data. 



By combining the capacitance vs. pressure calibration with the pressure vs. 
temperature relationship, we can convert capacitance into temperature. However, it is still 
necessary to obtain an absolute calibration of the temperature curve using fixed 
temperature points. In this experiment, Ta and TV, were used for this purpose. These 
fixed points, in the ^He phase diagram, can be easily identified as changes in slope when 
slowly (50 jiK/hr) sweeping temperature (see Figs. 2.16 A and B). 

The absolute calibration results in the y-axis of the temperature vs. capacitance 
curve being adjusted by a constant value to match the fixed points. During the two ^He 
experiments listed in this dissertation, the vertical adjustment was typically 0.5 |j,K. The 



43 



(0 
CO 



oo 


Jl 


1 1 — 


T — 1 1 1 1 1 1 1 r 


— 1 — 1 1 1 IT| ■ 1 


1 


34 


]1 


' ,1 


"^^^^^^^--^..^^^ 


Solid Phase 


- 




>l 


ABA 


^X 




- 


33 


— 








- 


32 


" 








■ 




- 




Liquid Phase 




■ 


31 


- 








- 


30 


- 








- 


OQ 


"ll 


1 ' 


1 1 1 1 1 1 1 1 1 


1 1 1 1 1 1 1 1 





10 



100 



T(mK) 



Figure 2.15: Melting curve of ^He as determined by W. Ni [59]. The superfluid ^He 
ordering transitions, A (2.505 mK), AB (1.948 mK), and the solid ordering transition N 
(0.934 mK), are marked with arrows. 



44 



31.760 
31.755 



f 31.750 

o 

c 

(0 

■^ 31.745 

CO 

Q. 
(Q 

O 31.740 



31.735 



31.730 



— 1 


1 1 1 1 1 1 1 1 1 1 1 


I 1 1 

A ^ 






- 


- 


r^ = 31.745 + 0.002 pF 


\. 






- 



13000 



14000 



15000 

Time (s) 



16000 



17000 



31.790 




31.780 



4000 



5000 6000 

Time (s) 



7000 



Figure 2.16: Identification of the capacitance value of the temperatures, Ta and TV, fi-om 
plots of the MCT capacitance vs. time while slowly warming. Solid lines have been added 
as guides to the eye. Both figures are different sections of the same data set. 



45 

final temperature vs. capacitance relationship is shown in Fig. 2.17. The solid line is a fit 
to a 1^ order polynomial. This polynomial was used to convert the MCT capacitance 
values into temperature above Tn. 

The Pt NMR thermometer was calibrated using the MCT by sweeping the 
temperature fi-om 3 mK to 0.5 mK and recording both the MCT and Pt NMR 
thermometer. There was a wait of at least 5 minutes between each Pt NMR pulse so that 
the nuclear spins could relax. The MCT temperature was recorded before and after each 
Pt NMR trace and averaged. The entire temperatiire sweep would take approximately 
12 hours. The temperature scale was checked by comparing Tc{P) [61] with the 
temperature where there was a dramatic crossover fi-om high to low attenuation in the 
zero sound signal. A graph of MCT temperature vs. Pt NMR integrated signal is shown in 
Fig. 2.18. The digitization rate of the TDS-430A oscilloscope was faster than the PLM-3, 
and therefore each Pt NMR trace was read by the oscilloscope and integrated after the 
experiment by the Labview software. The solid line represents a fit to the data fi-om 
Tn (0.934 mK) to 1 .5 mK using 



^MCT =~ T' (2.6) 

Mp, +B 



where Tmct is the MCT temperature, Mp, is the Pt NMR integrated signal, and ^ and 5 are 
fitting parameters. The temperature range for the fit was chosen so that the Pt NMR 
integrated signal was at least a factor of 10 above the noise. The ^He melting curve 
determined by W. Ni and co-workers [59] did not extend much above 322 mK (the 



46 

minimiiin in the melting curve). At this temperature, a separate calibration was used and 
was based on measurements by Grilly et al. [62]. The resultant temperature calibration 
from the Grilly et al. scale was adjusted by a constant to match the value of the ^He 
melting curve at the minimum given by Ni et al. [59]. 



4.5 
4.0 
3.5 
3.0 



^ 25 

H 2.0 



1.5 
1.0 
0.5 




o Temperature vs. Capacitance 
7th Order Polynomial Fit 

-J ^ I I I . I 



31.70 31.72 31.74 31.76 

Capacitance (pF) 



31.78 



Figure 2.17: The temperature vs. capacitance curve generated by combining the 
capacitance vs. pressure relationship in Fig. 2.14 with the ^He mehing curve in Fig. 2.15. 
The curve has been adjusted by a constant to match the fixed temperature points, Ta and 
Tn. The solid line is 7"' order polynomial fit. 



47 




O 0.6 - 



5 10 15 20 25 

Pt NMR Integrated Signal (a.u.) 



Figxire 2.18: The MCT temperature vs. Pt NMR integrated signal. The solid line is a fit 
to Eq. 2.6 fi-om Tn to 1.5 mK. 



CHAPTER 3 
THEORETICAL TECHNIQUES 

In this chapter, two theoretical models are presented that were used to investigate 
the magnetic behavior of the low dimensional magnetic materials reported in this 
dissertation. In each section, the relevant theory as weD as details of the software are 
discussed. In addition, the advantages and restrictions of each model are considered. The 
first section discusses the exact diagonalization method, which is essentially the calculation 
of the partition function for a cluster of spins. Although, this approach, in principle, is an 
exact calculation, the number of spins included with this technique is limited by the 
computing power. This restriction places limits on the temperature range over which 
these systems can be accurately modeled. With respect to the Hamiltonian, this method is 
flexible and can be applied to any cluster of interacting spins with only minor changes to 
the software. The second section considers the mapping of the ladder Hamiltonian onto 
the XXZ model. Although the XXZ model is exactly solvable, the mapping approximates 
the magnetic behavior since it only includes the low energy states of the ladder 
Hamiltonian. Nevertheless, this method can model the magnetic behavior of ladders at 
temperatures significantly below the thermal energy represented by the magnetic 
interactions. 



48 



49 



3.1 Exact Diagonalization 
The partition function for an arbitrary system of discrete states is written as 



Z=2]exp 






(3.1) 



where E, are the energy states of the system and Kb is the Boltzmann constant. For a 
cluster of spins, the magnetization can be calculated in a straightforward manner once the 
partition function is known; i.e. 



M 



i 


exp 




J^exp 




t] 





(3.2) 



where 5, represents the total spin of each state. A cluster of two Ising 5=1/2 spins that 
interact with a single exchange is the simplest cluster to model. This system can be 
represented by a Hamiltonian, where J is the magnetic exchange, given by 



(=1 ;=1.2 



(3.3) 



In Eq. 3.3, the symbol g represents the Lande g factor and //g is the Bohr magneton. The 
first and second terms represent the interaction between spins and the interaction with the 



50 



magnetic field, respectively. Using Eqs. 3.2 and 3.3, we arrive at the magnetization for n 
moles of spins that are arranged as interacting Ising pairs; namely 



M 



^"^Ag^B^ 



exp 






f_ 



exp 






exp 






+ exp 



kj 



+ 2 exp 



J 



\' 



y2kgT J 



(3.4) 



where A'^ is Avagadro's number. It is easy to see that this equation will behave properly in 
the limit of high temperature or field. 

We can use a similar method to calculate the magnetization for a system of 
Heisenberg spins. Again, for simplicity, we consider a system with only two interacting 
spins and a single exchange constant J. The Hamiltonian resembles Eq. 3.3, except that 
the spin operator is now a vector. 



^.s=^ZS,-S,.,-^/^.ZS,.H 



(3.5) 



'=1,2 



In the Ising case, the spin basis states were also the eigenvectors of the spin operator. For 
the Heisenberg case, this is not true. We explicitly write the spin basis eigenvectors that 
represent the electron wave functions with either spin up, |(x) , or spin down, |3) . The 

spin operator S, can be divided into its components S, = Sf + S] + Sf . The components 
are also operators that act on the spin basis function according to the rules given in 



Table 3.1. 



51 



Table 3.1: The spin operators acting on the spin basis functions. 





S.ol) 


•5,3) 


Sz 


Yi^) 


Yi^) 


Sx 


Yi^) 


Yi^) 


A 

Sy 


Yi^) 


'Yi^) 



We apply the Hamiltonian, Eq. 3.5, onto the basis function for two spins to obtain 
a matrix representation for the energy states of that system. Omitting the field interaction 
term, this matrix may be written as: 





aa) 


aJ3) 



15a) 



Pl^) 


{aa 


JIAr 







{Pa 







-J/4 
Jll 


Jll 
-JIA 


{PP 








J/4 



0. 



(3.6) 



The matrix is blocked according to the total spin value, S= 1,-5=0, and 5 = -1. Each 
matrix corresponding to a particular spin can be diagonalized individually. For the case of 
only two spins, it would be just as easy to diagonalize the entire matrix at once. However, 
for large clusters of spins, a great reduction in the necessary computing power is achieved 
by diagonalizing each total spin matrix individually. After determining the eigenvalues, the 



52 

field interaction term may be included. For two spins, we obtain the eigenvalues: -3J/4, 
J/4, JIA - gjusH, and J/4 + g^aH, which correspond to the familiar singlet and triplet states 
for two interacting .S" = 1/2 spins. The triplet states are degenerate in zero magnetic field, 
and we can display this graphically using the diagram sketched in Fig. 3.1. 




5^=1 



S = 



M, = l 



M,= 





Ms- 





M,= 


-1 



Figure 3.1: A graphical depiction of the energy eigenvalues for a system of two 5=1/2 
spins in a magnetic field showing the singlet and triplet states. 



Once we have the eigenvalues, it is trivial to plug them into Eq. 3.2 to obtain the 
magnetization. Again, for practical purposes, we have assumed n moles of spins arranged 
as dimer pairs, and have 



M = 



nN^gMa 



exp 






-exp 



-SMafi 



exp 






+ exp 






+ exp 






+ 1 



(3.7) 



53 



In the low field limit, Eq. 3.7 becomes the Bleaney-Blowers equation [42] ; which can be 



written as 



r^N.g'^il 



3kJ 



1 



l+^xp(j/A:,r) 



(3.8) 



At high temperatures, Eqs. 3.7 and 3.8 become the 5=1/2 Curie law, namely 



2 ..2 



z = 



nN.g^MB _nN,g'^ilS{S + X) 



AkJ 



3kJ 



(3.9) 



Figure 3.2 shows a graph of molar magnetic susceptibility vs. temperature in a 
magnetic field of 0.1 T produced using Eq. 3.7 with an exchange constant of 7= 12 K. 
Below the peak temperature of approximately 7 K, there is an exponential decrease in 
susceptibility due to singlet formation. The peak in the curve corresponds to the thermal 
energy (gap) needed to form triplets. This type of curve is common to low dimensional 
gapped magnetic systems. 

Using this approach, we could have calculated other thermodynamic quantities as 
well. For instance, using the partition function, Z, and the energy eigenvalues, £„ for two 
Heisenberg spins, we write the entropy [63] of this system as: 



54 



cr = — exp 
Z 



^ £. Y £. 



V *-B^y 



ykgT 



+ lnZ 



+— exp 
Z 



A/' 



V ^s^y 



^ + lnZ 



— exp 
Z 



V 



V "-B-* y 



ykgT 



+ lnZ 



y 



^kgl 



H — exp 
Z 



(3.10) 






y^kgl 



+ lnZ 



Figure 3.3 shows the entropy vs. temperature for a system of two Heisenberg spins that 
interact with an exchange constant J in a magnetic field of 0. 1 T. In the high temperature 
limit, the entropy approaches ln(number of energy states) = hi(4), and in the low 
temperature limit, it approaches zero. 



o 

E 



(U 

e 




Figure 3.2: The molar magnetic susceptibility vs. temperature in a magnetic field of 0.1 T 
for dimer pairs of Heisenberg spins that interact with an exchange constant of J= 12 K. 
The curve was produced using Eq. 3.7. 



55 




J=12K 



J I I 



40 



Figure 3.3: The entropy vs. temperature in a magnetic field of 0.1 T for a dimer pair of 
Heisenberg spins that interact with an exchange constant of J = 12 K. The curve was 
produced using Eq. 3.10. 



Calculating magnetization using the partition lunction is only trivial when the 
number of interacting spins, and hence the number of energy states, is small. This exact 
diagonalization method was used by Robert Weller [64] in 1980 to calculate the 
susceptibility for larger clusters (TV > 12) of 5' = 1/2 magnetic spins. This method relies on 
mathematics that have long been understood, however, it was not a viable alternative until 
cheap computing power was available. Although this method can be easily scaled up to 
larger systems, the corresponding matrix size increases as the factorial of the number of 



spins. For N spins, the maximum matrix size of 






, or A^ choose Nil, is such that 



when A^ = 12, the matrix size is 924 x 924. A matrix of this size can be manipulated by a 



56 

desktop computer in a few minutes. For 20 spins, the maximum matrix size 1 84,756 x 
1 84,756 and the computation quickly becomes impossible. At higher spin values, e.g. S = 
1, the matrix size increases even faster. However, we have not utilized all possible 
symmetries of the problem. By considering geometric symmetry of the spins, we can 
reduce the problem computationally by several orders of magnitude. This method is 
referred to as the Lanczos algorithm [65,66]. Using this approach, the magnetization for a 
system of as many as 30 spins can be calculated. I did not use this method, and so I do 
not discuss the details here. 

Since the importance of the boundaries of a model system decrease with increasing 
system size, it is important to use as many spins as possible. In addition, a small number 
of spins can accurately describe a low dimensional material only as long as the correlation 
length does not exceed the total length of the system. At 7= 0, for quasi-two dimensional 
systems, such as spin ladders, the correlation length can become infinitely long. For these 
reasons, this method will only give accurate results for temperatures that are the same 
order of magnitude as the exchange constants or higher, T>J. If the system size is too 
small, this method introduces erroneous plateaus in the magnetization curves as the 
temperature is lowered below the exchange constants. 

In this dissertation, the spins were arranged in either a ladder or ahemating chain 
geometry. However, any arrangement consisting of interacting spins with exchange 
constants, Ji, J:, ... Jn, could have been used. Both the ladder and alternating chain model 
systems used 12 spins which were arranged in a ring to help alleviate the boundary 
problem. The programs were written in matlab and produced theoretical curves using 
the exact diagonalization method described above. I am grateful to Steve Nagler (ORNL) 



57 

for his assistance in writing these programs, which have been included in Appendix H. 
Fitting the data involved three steps. First, the experimental curves were fit using a high 
order polynomial ( > 5). Second, the software would generate multiple curves over a 
preset parameter space. Each curve would be compared to the polynomial and the 
difiference between the polynomial curve and the theoretical curve would be recorded as a 
chi value. The chi value is the sum of the square of the difference between each 
theoretical point and the polynomial curve. Finally, when the program was finished 
generating curves, the chi^ values would be searched to obtain the lowest value, hopefully 
corresponding to the best fit. It was beneficial to generate a curve using those final 
parameters to ensure that the theoretical curve matched the data. A typical parameter 
search generates approximately 300 curves and takes approximately 18 hours. It is 
possible to increase the eflSciency of the process by allowing the software to choose the 
next parameters instead of blindly searching the whole parameter space. This procedure is 
described in the dissertation by Robert Weller [64]. However, this technique was 
abandoned as it tended to find local minima in the parameter space. 

3.2 The XXZ Model 
During my investigation of low dimensional materials, it became necessary to 
produce low temperature (T « the lowest ladder exchange constant, e.g. ^| « 4 K) 
magnetization curves for spins arranged in a ladder geometry. At the lowest experimental 
temperature of 0.7 K, there exists a feature in the data at half the saturation magnetization, 
Ms/2, that could not be modeled using the exact diagonalization method. As discussed in 
the previous section, the exact diagonalization procedure is increasingly inaccurate as the 



58 

temperature is lowered below the exchange constants. In addition, the exact 
diagonalization method also introduced erroneous plateaus in the theoretical curves that 
resemble the feature at Ms 12. Therefore, another method was required to study the 
magnetization of ladder materials at low temperature. 

Chaboussant et al. [14] have previously created low temperature magnetization 
curves for the ladder-like material, Cu(Hp)Cl, by mapping the ladder Hamiltonian onto the 
XXZ model which was initially solved by H. Bethe [21]. The thermodynamics of the 
XXZ model have been completely described by Takahashi and Suzuki [67]. I begin with 
the ladder Hamiltonian including the field interaction term 



NI2^ _ Af-2 

2-,^2/-l •^2, +'^||Z- 
1=1 1=1 



^Uer=^lZS,,_,.S„+y|,2S,-S„,+g//^|;S,.H . (3.11) 



We can consider only the restricted Hilbert space composed of a singlet \S = Q,ms=0) 
and the lowest energy triplet l^ = 1,^5 = -l) on each rvmg. These are the two lowest 
energy states (see Fig. 3.1) and therefore the most populated. This approximation is valid 
since we are interested in the critical region where the magnetic field is on the order of Jj.. 
We can rewrite the effective Hamiltonian on this restricted Hilbert space as 



^^ ^^iiZcs." -K.^K-K. ^\K ■K.)+Hj±sf , (3.12) 



r=l 



r=I 



where the effective field is given by 



59 



H,,=J,+^-gMsH , (3.13) 



and S^ now represents the total spin of rung r. It should be noticed that this Hamiltonian 
is completely symmetric around H^^ = . Hence, any quantities computed from this 
Hamiltonian will also be symmetric around this point. This Hamiltonian (Eq. 3.12) can be 
identified as the effective S = \/2 XXZ model. The thermodynamics of the XXZ model 
have been reduced to a set of non-linear differential equations by Takahashi and Suzuki 
[67]; such that 



In rjix) = -3 V3 ^— ^ + s{x) * ln(l + u(x)) , (3.14) 

kgT 



u(x) = 2K(x)cosh 



+ IC (x) , (3.15) 



V 2kgl J 



and lnA:(x) = ^(x)*ln(l + /7(x)), (3.16) 



where six) = — sec /? 
4 



— X 

v2 J 



* is the convolution product and ri(x), u(x), and k(x) are 



parameters in the model. These equations must be solved iteratively from a known 
solution for each value of the temperature and magnetic field. In this case, the known 
solution was tj{x) = 3 and k{x) = 2 for J|| = and H^^ = 0. The convolution products 

are calculated as discrete integrals using 200 points. Since, the hyperbolic secant fimction 

X 

and hence, s{x), decays quickly, — was used instead of x in the argument to increase the 



60 



resolution. The convolution product must therefore be divided by 10 as well. Typically 
10 iterations were sufficient to reach equilibrium with a 200 point resolution. Once a 
stable solution is reached, the free energy per spin can be calculated using 



- = -'^-kJ\nK{Q). (3.17) 

N 2 



dF 

The magnetization is proportional to M = . The curves produced must be 

dH 

normalized so that the maximum overall magnetization is 1. Curves generated this way 

using J|i = were compared to the exact dimer results, Eq. 3.7, to ensure that the method 

was correct. It should be noted that there are three typographical mistakes in the 

treatment of Chabbousant et al. [14]: a sign error and a missing s(x) factor in Eq. 3.14 (or 

Eq. 30 as listed in the Chaboussant et al. paper) and a factor of 2 difference in Eq. 3.17 

(or Eq. 32 as listed in the Chaboussant et al paper). These integral equations were solved 

using MATLAB Software. For reference, these programs are included in Appendix H. 



CHAPTER 4 
STRUCTURE AND CHARACTERIZATION OF A NOVEL MAGNETIC SPIN 

LADDER MATERIAL 



Magnetic spin ladders are a class of low dimensional materials with structural and 
physical properties between those of ID chains and 2D planes. In a spin ladder, the 
vertices possesses unpaired spins that interact along the legs via J\\ and along the rungs via 
Jx, but are isolated from equivalent sites on adjacent ladders, i.e. interladder J' «J\\, Jj_. 
Recently, a considerable amount of attention has been given to the theoretical and 
experimental investigation of spin ladder systems as a result of the observation that the 
microscopic mechanisms in these systems may be related to the ones governing high 
temperature superconductivity [2,6]. The phase diagram of the antiferromagnetic spin 
ladder in the presence of a magnetic field is particularly interesting. At T = with no 
external applied field, the ground state is a gapped, disordered quantum spin liquid. At a 
field Hch there is a transition to a gapless Luttinger liquid phase, with a fiirther transition 
at Hc2 to a frilly polarized state. Both Hci and Hc2 are quantum critical points [1]. Near 
Hci, the magnetization has been predicted to obey a universal scaling fiinction [68]. Using 
a symmetry argument, this universal scaling can also be shown to be valid at Hc2- Until 
now, this behavior has not been observed experimentally. 

A number of solid state materials have been proposed as examples of spin ladder 
systems, and an extensive set of experiments have been performed on the compound 



61 



62 



Cu2(l,4-diazacycloheptane)Cl4, Cu2(C5H,2N2)2Cl4, referred to as Cu(Hp)Cl [7]. The 
initial work identified this material as a two-leg 5=1/2 spin ladder [7-14]. Although 
quantum critical behavior has been preliminarily identified in this system near Hci, this 
assertion is based on the use of scaling parameters identified fi-om the experimental data 
rather than the ones predicted theoretically [13,14]. Furthermore, more recent work has 
debated the appropriate classification of the low temperature properties [15-19]. Clearly, 
additional physical systems are necessary to experimentally test the predictions of the 
various theoretical treatments of two-leg 5=1/2 spin ladders. 

Herein, we report evidence that identifies bis(piperidinium)tetrabromocuprate(II), 
(C5Hi2N)2CuBr4 [20,69], hereafter referred to as BPCB, as a two-leg S = Ml ladder that 
exists in the strong coupling limit, JJJ\\ > 1. High-field, low-temperature magnetization, 
M{H < 30 T, T > 0.7 K), data of single crystals and powder samples have been fit to 
obtain Jx = 13.3 K, Jn = 3.8 K, and A ~ 9.5 K, i.e. at the lowest temperatures finite 
magnetization appears at Ha = 6.6 T and saturation is achieved at Hc2 = 14.6 T. An 
unambiguous inflection point in the magnetization, M{H,T =0.7 K), and its derivative, 
dMIdH, is observed at half the saturation magnetization, Ms/2. This behavior has not been 
detected in Cu(Hp)Cl [8-10]. The Ms/2 feature cannot be explained by the presence of 
additional exchange interactions, e.g. diagonal fi-ustration Jf, but is well described by an 
effective XXZ chain, onto which the original spin ladder model (for strong coupling) can 
be mapped in the gapless regime Hci < H < Hc2- After determining Hci and with no 
additional adjustable parameters, the magnetization data are observed to obey a universal 



63 



scaling function [68]. This observation further supports our identification of BPCB as a 
two-leg 5=1/2 Heisenberg spin ladder with J' « J\\. 

This chapter is divided into six sections. In the first section of this chapter, I vdll 
discuss the structure and synthesis of BPCB. The second and third sections report the 
results of low field susceptibility and magnetization measurements, respectively. The 
fourth section presents the high-field magnetization work performed at the National High 
Magnetic Field Laboratory, while section five details the universal scaling behavior of 
BPCB. Results fi-om the neutron scattering experiments, performed at Oak Ridge 
National Laboratory, are provided in section six. 



4.1 Structure and Synthesis of BPCB 
The crystal structure of BPCB has been determined to be monoclinic with stacked 
pairs of 5 = 1/2 Cu^^ ions forming magnetic dimer units [20]. The CuBr4"^ tetrahedra are 
co-crystallized along with the organic piperidinium cations so that the crystal structure 
resembles a two-leg ladder. Fig. 4.1. The rungs of the ladder are formed along the c*-axis 
(the c*-axis makes an angle of 23.4° with the a-c plane and the projection of the c*-axis in 
the a-c plane makes an angle of 19.8° with the c-axis) by adjacent flattened CuBr4 
tetrahedra related by a center of inversion. The ladder extends along the a-axis with 
6.934 A between Cu^"^ spins on the same rung and 8.597 A between rungs. The three 
dimensional crystal structure of BPCB, including the organic cations, viewed along the 
c-axis is shown in Fig. 4.2. The atomic positions have been taken fi-om the x-ray 



■2 



64 




Figure 4.1: A schematic diagram of the crystal structure of BPCB viewed down the [010] 
axis as determined by Patyal et al. [20]. The magnetic exchange between .S = 1/2 Cu ^ 
spins is mediated by non-bonding Br-Br contact. The two primary exchange models 
considered were a ladder model, with parameters Ji and J\\, and alternating chain model, 
with parameters J\ and J2. In the ladder model it is possible to include a frustration 
exchange, Jf- 



65 





Cu 



Br 



O c 



O H 



Figure 4.2: The crystal structure of BPCB viewed along the c -axis. The ladder direction 
is along the a-axis. The c*-axis, rung direction, makes an angle of 23.4° with the a-c plane 
and the projection of the c*-axis in the a-c plane makes an angle of 19.8° with the c-axis. 
The solid lines indicate the unit cell. 



66 



scattering data of Patyal et al. [20] and verified in the neutron scattering studies (see 
Section 4.6). The hydrogen positions have been calculated using symmetry arguments. 
The ladder structure is viewed edgewise (dark spheres) in Fig. 4.2, and it is apparent that 
the rungs of the ladder extend out of the a-c plane. Adjacent ladders are separated by 
12.380 A along the c-axis and 8.613 A along the b-axis. Although the b-axis separation is 
approximately the same as the rung separation, it is unlikely that the organic cations 
provide significant superexchange between ladders along the b-axis and hence the 
magnetic exchange between ladders is expected to be small {J' « J\\). The magnetic 
exchange, Jx, between Cu^^ spins on the same rung is mediated by the orbital overlap of 
Br ions on adjacent Cu sites. The exchange between the legs of the ladder, J\\, is also 
mediated by somewhat longer non-bonding (Br • • • Br) contacts and possibly augmented 
by hydrogen bonds to the organic cations. A diagonal exchange, Jf, is possible, although 
it should be weak (Jf « -^h), and since the diagonal distances (9.918 A vs. 12.066 A) are 
not equal, only one Jf was considered in our analysis. 

Shiny, black crystals of BPCB were prepared by slow evaporation of solvent from 
a methanol solution of [(pipdH)Br] and CuBr2, and milling of the smallest crystals was 
used to produce the powder samples. The stochiometry was verified using carbon- 
hydrogen-nitrogen analysis [70]. In addition, deuterated single crystal and powder 
samples were produced and used in neutron scattering studies performed at the High Flux 
Isotope Reactor at Oak Ridge National Laboratory. The protonated BPCB material has a 
molecular weight of 583.49 g/mol and a density of 2.07 g/cm^ 



67 



The previous study by Patyal et al. [20] reported the Lande g factor along all three 
single crystal axes for BPCB as g(a-axis) = 2.063, g(b-axis) = 2.188, and g(c-axis) = 
2.148. ESR measurements at a frequency of 9.272 GHz were performed on a powder 
sample of BPCB at room temperature and on a single crystal sample along the c-axis from 
20 to 300 K. The room temperature results were completely consistent with the 
previously reported data, i.e. g(powder) = 2.13 [20] and g(c-axis) = 2.148. At all 
temperatures, the EPR signal consisted of a single broad line approximately 500 G wide. 
Figure 4.3 is a sample derivative trace, d//d// of the EPR signal intensity at 75 K. By 
plotting the area under the EPR intensity curve I{H) as a fiinction of temperatxire, we 
obtain the graph shown in Fig. 4.4. This graph closely resembles the susceptibility curve 
of BPCB. The Lande g factor measured along the c-axis decreases monotonically from 
2.148 to 2.141 from 300 K to 20 K as shown in Fig. 4.5. This magnitude of change in the 
Lande g factor will not adversely affect the quality of the magnetization fits, which 
assumed g to be the temperature independent value of 2.148 along the c-axis. 

4.2 Low Field Susceptibility Measurements 
Although the crystal structure of BPCB resembles a ladder, other possible 
exchange pathways can produce similar results from macroscopic measurements [71,72]. 
Initially, an additional magnetic exchange model, i.e. alternating chain, was considered 
during the analysis of the magnetization data. Figure 4.1 shows the two primary exchange 
pathways considered, i.e. an alternating chain with exchange constants J\ and J2, and a 
ladder with exchange constants Jj. and J\\. The Hamiltonians for A^ spins that interact with 



68 



20 


, 1 1 1 1 — 1 1 T 

A 7=75K ; 


15 


/ \ m= 18.6 mg - 




r / \ /-/ II c-axis 


10 


/ \ ■ 


6 5 


/ 1 ; 


i: 


'——^^^^ \ 


- \ /-^^ ; 


-10 


\ / '■ 


-15 


— I— ^ 


-20 


1 . 1 — , — 1 — , — 1 — . — 



2000 2500 3000 3500 

Field (G) 



4000 



4500 



Figure 4.3: The first derivative of the EPR signal intensity, dlIdH, vs. field for a BPCB 
single crystal {m = 18.6 mg) at a fi-equency of 9.272 MHz and 75 K. 




300 



Figure 4.4: The integrated EPR signal intensity vs. temperature for a BPCB single crystal 
(w = 18.6 mg) at a fi-equency of 9.272 MHz. 



69 



2.150 



2.148 



-2 2.146 

^ 2.144 

c 

(0 



2.142 



2.140 



m = 18.6 mg 
H II c-axis 




I L 



50 100 150 200 250 300 

T(K) 

Figure 4.5: The Lande g factor along the c-axis of a single crystal sample of BPCB 
(w=18.6 mg), determined by the EPR line center frequency at 9.272 MHz, vs. 
temperature. The room temperature value of g agrees with the value reported earlier [20] 
{g = 2.148.). The temperature dependence of the Lande g factor is most likely due to the 
thermal contraction of the lattice. 



either a ladder or alternating chain exchange can be written as 



N/2 



N-2 



^der =^lZS„_, .S„ +/„XS, -S,,, +g//,XS, -H (4.1) 

;=1 /=l (=1 



and 



NI2 



NI2-\ 



9^ =^.ZS„_, -S,, +/, Y}„ -S,,,, +g/.,XS, -H , 



1=1 



;=1 



i=\ 



(4.2) 



respectively. 



70 



Low field (// < 5 T) magnetic measurements were performed using a Quantum 
Design squid Magnetometer. The low field, 0.1 T, magnetic susceptibility, x, of a BPCB 
powder sample (w = 166.7 mg) is shown as a fimction of temperature in Fig. 4.6. The 
general shape of the curve is typical of low dimensional magnetic systems, and more 
specifically, it possesses a rounded peak at approximately 8 K and an exponential 
dependence below the peak temperature. No evidence of long range order was observed 
at the minimum temperature of 2 K. A temperature independent diamagnetic contribution 
of Xdiam = -2.84 X 10~* cmu/mol was subtracted from the data in Fig. 4.6. The 
diamagnetic contribution is the sum of the core diamagnetism, estimated fi-om Pascal's 
constants to be -2.64 x 10^ emu/mol, and the background contribution of the sample 
holder. For all of the susceptibility data in this chapter, a diamagnetic contribution has 
been subtracted fi-om the data and although no Curie impurity term was subtracted, in 
some cases a 5 = 1/2 Curie contribution was included in the fit. This Curie contribution is 
typically ~2 % of the total number ofS= 1/2 spins. In Fig. 4.6, the susceptibility data 
have been fit (solid line) using a high temperature expansion by Weihong et al. [11] based 
on the ladder Hamiltonian, Eq. 4.1, providing the exchange constants of Ji = 13.1 ± 0.2 K 
and J|| = 4.1 ± 0.3 K. The first 14 terms of the expansion (up to fourth order in JJT) 
were used for the fitting procedure. These same data were also fit (Fig. 4.7) using the 
method of Chiara et al. [7], which assumes the alternating chain Hamiltonian, Eq. 4.2, 
providing the exchange constants of Ji = 13.74 + 0.03 K and J2 = 5.31 ± 0.04 K. The 
fitting method of Chiara et al. [7] includes the data below the peak in the susceptibility and 
consequently is more accurate than the high temperature series expansion method of 



71 



Weihong et al. [11]. However, although there are differences in the values of the two sets 
of exchange parameters, both cases provide physically plausible results. Therefore, using 
only the low field x(T) data, we are unable to distinguish between the ladder and 
alternating chain model. Similar results are obtained for BPCB single crystal samples. 
The magnetic susceptibility vs. temperature for BPCB single crystal (m = 46.9 mg) is 
shown in Fig. 4.8. The sample was zero field cooled to 2 K and then measured in a field 
of 0.1 T parallel to the a-axis. A small constant diamagnetic contribution of Xdiam = 
-3.16 X 10"* emu/mol has been subtracted. Incidentally, the diamagnetic contribution for 
the single crystal samples is larger because more diamagnetic support material was used to 
ensure proper crystal alignment during the measurement. The solid line represents the 
best fit using a high temperature series expansion by Weihong et al. [11] with the 
parameters Ji = 12.9 ± 0.3 K and J|| = 3.8 ± 0.3 K. Figure 4.9 shows this same data fit 
using the method of Chiara et al. [7] with the results Ji = 13.66 ± 0.14 K and Ji = 5.57 ± 
0.12 K. Analogous to the ladder and alternating chain fits of the powder susceptibility 
data, both fitting methods generate plausible results. In addition, although the exchange 
constants fi-om the single crystal and powder samples do not quite agree within 
uncertainty, the fitting results appear to be self consistent for both methods. The fitting 
results for powder and single crystal samples along all three axes have been summarized in 
Tables 4.1 and 4.2. 

The choice of 0.1 T as the applied field in the susceptibility measurements was not 
arbitrary. Figure 4.10 shows the molar magnetic susceptibility for BPCB single crystal 
with H II a-axis and applied fields of 1 , 2, 3, 4, and 5 T. At high temperatures (T > A/Icb), 



72 



25 



20 - 



O 

E 

"B 
E 

(D 

CO 

o 



15 



10 



.8 

O 
O 

o 

-O 

-o 
o 
o 
K) 
o 

B 



O Experimental Data 
Ladder Fit: 

J^ = 13.1 +0.2K 

J|i= 4.1+ 0.3 K 

Impurity Cone. = 1.2 + 0.2% - 




BPCB Powder 
H = 0.1T 



20 



40 60 

T(K) 



80 



100 



Figure 4.6: The molar magnetic susceptibility vs. temperature for BPCB powder 
(m = 166.7 mg). The sample was zero field cooled to 2 K and then measured in a field of 
0.1 T. A small constant diamagnetic contribution of Xdiam = -2.84 x 10"* emu/mol has 
been subtracted. The solid line represents the best fit using a high temperature series 
expansion by Weihong et al. [1 1] with parameters Jx, = 13.1 ± 0.2 K and J\\ = 4.1 ± 0.3 K. 



73 



25 



20 - 



O 
£ 15 

E 

? 10 



Experimental Data 
■Alternating Chain Fit: 
J, = 13.74 ± 0.03 K 
J^ = 5.31 ± 0.04 K 
Impurity Cone. = 0.9 ± 0.1 % 




BPCB Powder 
H = 0.1 T 



20 



40 



60 



80 



100 



T(K) 



Figure 4.7: The molar magnetic susceptibility vs. temperature for BPCB powder 
(m = 166.7 mg). The sample was zero field cooled to 2 K and then measured in a field of 
0.1 T. A small constant diamagnetic contribution of Xdiam = -2.84 x 10"^ emu/mol has 
been subtracted. The solid line represents the best fit using the method of Chiara et al. [7] 
with parameters J, = 13.74 ± 0.03 K and J2 = 5.31 ± 0.04 K. 



74 



25 



O 



Experimental Data 
Ladder Fit using: 
= 12.9 + 0.3 K 
J|i= 3.8 + 0.3 K 
Impurity Cone. = 3.3 + 0.4 % 




BPCB Single Crystal 
H = 0.1 T||a-axis 

I i_ 



20 40 60 

T(K) 



80 



100 



Figure 4.8: The molar magnetic susceptibility vs. temperature for a BPCB single crystal 
(m = 46.9 mg). The sample was zero field cooled to 2 K and then measured in a field of 
0.1 T parallel to the a-axis. A small constant diamagnetic contribution of Xdiam "= 
-3.16 X 10^ emu/mol has been subtracted. The solid line represents the best fit using a 
high temperature series expansion by Weihong et al. [11] with parameters A - 
12.9 + 0.3 K and ^, = 3.8 + 0.3 K. 



75 



25 



O 

E 

'3 
E 

CO 

o 



1 1 1 ' 1 ' 

Experimental Data 
-Alternating Chain Fit: 
= 13.66 ± 0.14 K 
= 5.57 ± 0.12 K 
Impurity Cone. = 3.3 ± 0.4 % J 




BPCB Single Crystal 
H = 0.1 T II a-axis 



_L 



_L 



_L 



20 



40 60 

T(K) 



80 



100 



Figure 4.9: The molar magnetic susceptibility vs. temperature for a BPCB single crystal 
(/w = 46.9 mg). The sample was zero field cooled to 2 K and then measured in a field of 
0.1 T. A small constant diamagnetic contribution of Xdiam = -3.16 x 10^ emu/mol has 
been subtracted. The solid line represents the best fit using the method of Chiara et al. [7] 
with parameters J, - 13.66 ± 0.14 K and J2 = 5.57 ± 0.12 K. 



76 



O 

E 

"3 
E 

(D 

CO 

b 



25 



20 - 



15 - 



10 - 



5 - 



1 


1 1 I 1 


— 1 — 1 — 1 — 


' 1 ' ' 


1 1 1 


-1 1 1 I 


. 




r-i (~t 






- 


. 




Q.^-^~ 


"•^Sd^ 




- 


_ 




r 


^^5^ 




- 


_ 




/ 


^CK 


t*v 


— 


. 




f 




^5^ 


- 


_ 








^D^ 


"vv 


- 










^5-^ ■ 


- 










^^ 


7 


D^ 








■tI 


_ 


D^ 








- 


. 


D^ 








- 


- 


D^ 








- 1 Tesia - 


: 


0^ 









2 TesIa - 


: 


nW 






A 


3 TesIa - 


'_ 








V 


4 TesIa i 


- 








D 


5 TesIa \ 


_ 




H|| 


a-axis 




■ 




, , , 1 


1 1 1 


1 1 1 1 


1 1 1 


1 1 1 1 



10 

T(K) 



15 



20 



Figure 4.10: The molar magnetic susceptibility vs. temperature for a BPCB single crystal 
(w = 14.2 mg). The sample was zero field cooled to 2 K and then measured in the fields 



of 1, 2, 3, 4 and 5 T. A small constant diamagnetic contribution of -4.06 x 10 emu/mol 
has been subtracted. The data collapse onto a single curve at high temperatures 
(r> A/Icb) indicating the approximately constant susceptibility at high temperature. 



77 



Table 4.1: The alternating chain parameters, J\ and Ji, determined from fitting the 
susceptibility vs. temperature data using the method of Chiara et al. [7]. 





mass (mg) 


J,(K) 


^2(K) 


Impurity Cone. (%) 


powder 


166.7 


13.74 ±0.03 


5.31 ±0.04 


0.9 ±0.1 


a-axis 


46.9 


13.66 ±0.14 


5.57 ±0.12 


3.3 ±0.4 


b-axis 


13.6 


12.76 ±0.10 


5.18±0.10 


4.8 ±0.5 


c-axis 


24.4 


13.65 ±0.10 


6.05 ±0.10 


3.8 ±0.4 



Table 4.2: The ladder parameters, Jx and J\\, determined from fitting the susceptibility vs. 
temperature data using the high temperature expansion by Weihong et al. [11]. 





mass (mg) 


Jx(K) 


^11 (K) 


Impurity Cone. (%) 


powder 


166.7 


13.1 ±0.2 


4.1 ±0.3 


1.2 ±0.2 


a-axis 


46.9 


12.9 ±0.3 


3.8 ±0.3 


3.3 ±0.4 


b-axis 


13.6 


13.4 ±0.3 


3.7 ±0.2 


7.5 ±1.0 


c-axis 


24.4 


13.3 ±0.4 


3.8 ±0.5 


3.0 ±1.0 



all of the susceptibility data collapse onto a single curve demonstrating the approximately 
constant susceptibility. However, below the peak, the susceptibility curves begin to 
deviate. In addition, the peak temperature decreases with increasing field. At fields above 
the gap, A/gi^B ~ 6.8 T, the peak in the susceptibility curve should disappear entirely. 
Although, a larger applied field would increase the signal to noise ratio of our 
measurements, we would measure the field and temperature dependence of the sample 
simultaneously, thus complicating our analysis. 

The inverse susceptibility as a Sanction of temperature for BPCB powder 
(/w= 166.7 mg) is shown in Fig. 4.11, and similarly, the inverse susceptibility vs. 
temperature for a BPCB single crystal with H \\ a-axis {m = 46.9 mg) is shown in 



78 



Fig. 4.12. At temperatures above the spin gap, T » A ~ 8 K, the inverse molar 
susceptibility should be linear with temperature. The slope of this line can be determined 
by inverting the 5=1/2 Curie law, 






(4.3) 



where Nk is Avagadro's number. The value of is somewhat more difficult to calculate. 
Johnston et al. [41] have written a high temperature series expansion, by inverting a 
susceptibility expansion from Weihong et al. [11], for the inverse susceptibiHty in terms of 
Jl and ^1 containing 42 non-zero terms. The first four terms of that series can be written 
as 



1 _ AkgT 



X N^n,' 



l + (2J,,+jJ^ + (2V+j/)^ + (2J,>J,^)^ + . 



(4.4) 



where x = — . By comparing Eqs. 4.3 and 4.4, we can write the Curie temperature, 0, 



to second order in T as 



= 



(2J|,+jJ ^ (2J,|^+j/) ^ (2V+J,^) ^ 



8r 



2Ar 



(4.5) 



79 



3 

E 

0) 

CO 

O 

o 

E 



0.8 
0.7 
0.6 
0.5 
0.4 
0.3 
0.2 
0.1 
0.0 



-1 1 1 1 1 1 1 1 1 — 

o Experimental Data 
Linear Fit (100 K to 300 K) 

slope = 2.313 ± 0.002 (mol/K emu) 

© = 4.9±0.3K 




BPCB Powder 
H = 0.1 T 
I I 



300 



Figure 4.11: The inverse molar magnetic susceptibility vs. temperature for BPCB powder 
(m = 166.7 mg). The sample was zero field cooled to 2 K and then measured in a field of 
0.1 T. A small constant diamagnetic contribution of Xdiam = -2.84 x 10 emu/mol has 
been subtracted. The solid line is a linear fit over the temperature range fi-om 100 K to 
300 K giving a slope of 2.3 13 ± 0.002 mol/(emu K) and = 4.9 ± 0.3 K. 



80 



Z2 

E 

(D 

n 

b 

"o 

E 



0.8 

0.7 

0.6 h 

0.5 

0.4 

0.3 

0.2 

0.1 

0.0 



"I 1 1 ' f 

o Experimental Data 



Linear Fit (100 K to 300 K) 
slope = 2.50 ± 0.01 (mol/emu K) 
= 5.8±1.2K 




BPCB Single Crystal 
H = 0.1 Tlla-axis 



50 100 150 200 

T(K) 



250 



300 



Figure 4.12: The inverse molar magnetic susceptibility vs. temperature for a BPCB single 
crystal {m = 46.9 mg). The sample was zero field cooled to 2 K and then measured in a 
field of 0.1 T parallel to the a-axis. A small constant diamagnetic contribution of Xdiam = 
-3.16x10"^ emu/mol has been subtracted. The solid line is a linear fit over the 
temperature range fi-om 100 K to 300 K with a slope of 2.50 ± 0.01 mol/(emu K) and = 
5.8 + 1.2 K. 



81 



After plugging in nominal values for our exchange constants and comparing the magnitude 
of each term, it is clear that we only need to consider the first term as long as our linear fit 
begins above approximately 100 K. The solid lines in Figs. 4. 11 and 4.12 represent linear 
fits over the temperature region fi-om 100 K to 300 K. The slope and Curie constant for 
these two samples, as well as BPCB single crystal specimens along the b and c-axis, are 
listed in Table 4.3 and compared to the theoretical result fi-om Eq. 4.5. The exchange 
constants, A and ^ , used in Eq. 4.5 have been taken from Table 4.2. 



Table 4.3: The slope of the inverse molar susceptibility, 1/x, vs. temperature and the 
Curie temperature, 0, determined from a linear fit to the data from 100 K to 300 K. The 
theoretical slope, Aknlgni^A (Eq. 4.3) and Curie constant, (Jx + 2Jii)/4 (Eq. 4.5), are 
included for comparison. The parameters used in the calculation of the Curie constant 
were taken from Table 4.2. 





slope 
(mol/K emu) 


(mol/K emu) 


0(K) 


(Jx + 2J||)/4(K) 


powder 


2.312 ±0.002 


2.35 ± 0.02 


4.9 ±0.3 


5.3 ±0.3 


a-axis 


2.50 ±0.01 


2.51 ±0.02 


5.8 ±1.2 


5.1 ±0.4 


b-axis 


2.24 ± 0.02 


2.23 ± 0.02 


3.9±1.1 


5.2 ±0.3 


c-axis 


2.29 ± 0.02 


2.31 ±0.02 


4.6 ±1.6 


5.2 ±0.6 



The molar magnetic susceptibility multiplied by temperature vs. temperature for 
BPCB powder {m = 166.7 mg) is shown in Fig. 4.13. The solid line is the theoretical high 
temperature Curie value for j^T of 0.425 (emu K)/mol. At temperatures above 
approximately 1 00 K, xT approaches a horizontal line with the Curie value, indicating 
paramagnetic behavior. Below 1 00 K, the value of zT decreases when lowering the 



82 



temperature, indicating antiferromagnetic behavior. If we had not subtracted the 
appropriate diamagnetic contribution, x^ ^t high temperatures would have a non-zero 
slope. 



0.5 



0.1 



0.0 



N,g\'l4k^ 




.ooooouuoo 



oooooooooooo 



BPCB Powder 
H = O.M 



J_ 



50 



100 



150 
T(K) 



200 250 



300 



Figure 4.13: The molar magnetic susceptibility multiplied by temperature vs. temperature 
for BPCB powder (w = 166.7 mg). The sample was zero field cooled to 2 K and then 
measured in a field of 0.1 T. A small constant diamagnetic contribution of 
Xdiam = -2.84 X 10 " emu/mol has been subtracted. The solid line is the theoretical Curie 
value of 0.425 (emu K)/mol. 



4.3 Low Field Magnetization Measurements 
The magnetization measurements were performed with a commercial SQUID 
magnetometer, which can apply a maximum field of 5.0 T. This limitation is particularly 
unfortunate in the case of BPCB, since the spin gap, expected fi-om the susceptibility 
measurements, is approximately 7 T. The spin gap is calculated to first order as 



83 



A/kB =Jl-J\\ (4-6) 

for the ladder exchange or 

A/kB=Ji-/2 (4.7) 

when considering an alternating chain model [41]. For each measurement listed, the 
samples were zero field cooled fi-om 300 K. The overall form of the low field 
magnetization measurements can be understood by examining the behavior of two 
Heisenberg 5=1/2 spins with a single exchange constant J. The molar magnetization of 
such a system can be calculated using Eq. 3.7. At low temperatures (T « A/ks), the 
magnetization will remain zero until the gap field is reached {H = A/gjus) and then 
afterwards have a positive first derivative. At high temperatures {T > H and T » J), 
Eq. 3.7 becomes approximately linear with applied field. Between these two temperature 
extremes, the magnetization will have a small positive first derivative (compared to the 
paramagnetic result ofdM{H)/dH < NAgHBl4kBT) and a positive second derivative. The 
molar magnetization vs. field for BPCB powder (jn =166.7 mg) at a temperature of 2 K is 
shown in Fig. 4.14. The solid line is a fit using Eq. 3.7 with an exchange constant of J = 
12.6 ±0.1 K. The general shape of the curve matches the data commendably considering 
the simplicity of the model. This agreement is an indication that, regardless of which 
magnetic model is correct, BPCB exists in the strongly coupled limit, i.e. JJJ\\ »1 or 

J,/J2»l. 



84 



To facilitate fitting the magnetization data more accurately, we used the exact 
diagonalization technique discussed in Chapter 3. For all of the fits that are discussed, 
unless otherwise noted, the calculations used 12 spins arranged in a ring. The molar 
magnetization as a fimction of field for a BPCB single crystal (m = 166.7 mg) at the 
temperatures of 2, 5, and 8 K is shown in Fig 4.15. The solid line at 2 K represents the 
best fit using the 12 spin exact diagonalization procedure and an alternating chain 
Hamiltonian, Eq. 4.2. The magnetization curves at 5 K and 8 K were produced using the 
same best fit exchange parameters derived fi-om the 2 K data, Ji = 13.20 + 0.05 K and Ji = 
5.20 ± 0.05 K. The experimental curve at 2 K is reproduced extremely well by this fitting 
technique. However, at higher temperatures, using the same exchange constants, the 
theoretical and experimental curves begin to deviate. The same fitting procedure can be 
applied to the data using the ladder Hamiltonian, Eq. 4.1. Figure 4.16 shows the same 
data fit using the exact diagonalization with a ladder Hamiltonian. The ladder best fit 
exchange parameters are Jj_ = 12.75 ± 0.05 K and ^ = 3.80 ± 0.05 K. Contrary to the 
case for the alternating chain Hamiltonian, the higher temperature experimental and 
theoretical magnetization curves agree using the same exchange constants at higher 
temperatures. This agreement suggests that the data may be more accurately modeled 
using the ladder Hamiltonian. 

There are two reasons why the error in fitting the low field magnetization 
measurements is relatively large compared to the error in fitting the susceptibility 
measurements. First, a small discrepancy in the mass or temperature measurement will 



85 



400 



300 
O 

E 
O 

33 200 

E 



100 



-< 1 ' 1 ' r 

BPCB Powder 
7 = 2K 

O Experimental Points 

Dimer Fit 

J= 12.6 + 0.1 K 




H(T) 



Figure 4.14: The molar magnetization vs. field for BPCB powder (m = 166.7 mg) at a 
temperature of 2 K. The solid line represents the best fit to Eq. 3.7, the molar 
magnetization for pairs of 5 = 1/2 Heisenberg spins with a single exchange constant of J = 
12.6 + O.IK. 



86 



O 

E 

c5 

E 

0) 



1000 



800 



600 



400 



200 



o 


7 = 


:2K 


A 


r= 


= 5K 


V 


7 = 


:8K 




bXdui L^iciyuiiciii^ciuuii 




J.-- 


= 13.20 K 




J2- 


= 5.20 K ^ 




H II a-axis 



H(T) 

Figure 4.15: The molar magnetization vs. field for a BPCB single crystal (m = 166.7 mg) 
with H II a-axis at the temperatures of 2, 5, and 8 K. The solid lines are produced using 
the best fit parameters, J, = 13.20 ± 0.05 K and J2 = 5.20 ± 0.05 K, determined fi-om the 
2K data using the 12 spin exact diagonalization procedure and an ahemating chain 
Hamiltonian. At higher temperatiires, using the same exchange constants, the theoretical 
and experimental curves begin to deviate. 



87 



1000 



o 
E 

S 

E 

0) 



800 



600 



400 



200 



O 
A 

V 



T=2K 

r=5K 

r=8K 

Exact Diagonalization 

J^= 12.75 K 

J= 3.80 K 




Figure 4.16: The molar magnetization vs. field for a BPCB single crystal (m = 166.7 mg) 
with H II a-axis at the temperatures of 2, 5, and 8 K. The solid lines represent the best fit 
to the 2 K data, using the 12 spin exact diagonalization procedure and a ladder 
Hamiltonian. The ladder best fit exchange parameters are Jx = 12.75 ± 0.05 K and ^j = 
3.80 ± 0.05 K. The higher temperature experimental and theoretical magnetization curves 
agree using the same exchange constants. 



88 



result in a large difference in the best fit exchange constants. Experimentally, it is easier to 
hold the magnetic field constant than the temperature. Second, because we did not reach 
the saturation magnetization, or even the critical field Hci, absolute calibration of the mass 
or the critical fields, Hci and Hci, is not possible. Determination of the exchange 
constants fi-om the low field magnetization data relies on the absolute magnetization 
values. On the other hand, the susceptibility data contains a maximum with a unique 
temperature dependence that is sensitive to the values of the exchange constants. Figure 
4.17 shows the 2 K magnetization data fi-om the previous two figures. The solid and 
dotted lines represent the exact diagonalization fits extended to 20 T using the alternating 
chain and ladder Hamiltonians, respectively. At a temperature of 2 K, it should be 
possible to distinguish between the two models by continuing the magnetization 
measurements to high field. By lowering the temperature to 1 K, this difference will 
become more pronounced (see Fig. 4.18). Figure 4.19 shows the magnetization data for 
Cu(Hp)Cl at 0.42 K [14]. The solid line represents the best fit using the exact 
diagonalization procedure and an alternating chain Hamiltonian with exchange constants 
Ji = 13.20 ± 0.05 K and J2 = 2.3 ± 0.05 K. The first derivative of the data and theoretical 
prediction are provided in the inset. The asymmetry of the curve is obvious from Fig. 4.19 
and is a result of the asymmetry in the magnetic exchange (see Fig. 4.19), i.e. J\ * Ji. 
These results suggest that Cu(Hp)Cl, which has been considered a two-leg ladder material 
[14], is better described by an alternating chain model. 



89 



6000 



5000 



O 4000 



E 

(D 



3000 



2000 



1000 



-| — 1 — I — I — I — I — r- 
Experimental Data 

- Exact Diagonalization using 
Alt. Chain Hamiltonian 

= 13.20 K 
J^ = 5.20 K 

- Exact Diagonalization using 
Ladder Hamiltonian 
J =12.75 K 
J.. = 3.80 K 




Figure 4.17: The molar magnetization vs. field for a BPCB single crystal vs. field fi-om 
to 5 T. The solid and dotted lines represent the exact diagonalization fits extended to 20 
T using the alternating chain and ladder Hamiltonians, respectively. At a temperature of 
2 K, it should be possible to distinguish between the two models by continuing the 
magnetization measurements to high field. 



90 



6000 



5000 - 



O 4000 

E 

O 

3 3000 

E 

^2000 



1000 




' 


-1 — 1 — 1 — 1 — 1 — 1 — 1 — \ — 1 — 1 — 1 — r 


1 1 ■ 1 > 1 


— 


■ 






Alt. Chain Hamiltonian 


// 






J, = 13.20 K , 


f / 




- 


Jj = 5.20 K / 


/ 




— 






Ladder Hamiltonian // 






■ 


J = 12.75 K // 






: 


J„ = 3.80 K / 

It X 




. 


— 






^ 


- 




H II a-axis 


- 


- 


I ■ ,.A< 1 ■ 1 1 1 1 1 


7=1 K 
1 1 1 1 1 





10 

Field (T) 



15 



20 



Figure 4.18: The solid and dotted lines represent the exact diagonalization fits fi-om the 
previous graph (Fig. 4. 1 7) calculated at a temperature of 1 K. At this temperature, the 
difference between the curves becomes more pronounced. At a field of approximately 
10.6 T, there appears to be an inflection in the predicted magnetization using the ladder 
Hamiltonian. 



91 



w 



CuHpCL 
7= 0.42 K 
o Data 



OO (D 



— 1 — r — I — I — I — r- 




Figure 4.19: The data from Fig. 7 in Reference 14 have been digitized. After 
interpolating to 200 equally spaced points, the 1st derivative was taken using 13 point 
smoothing (open circles in the inset). The theoretical curve was created with the exact 
diagonalization procedure using 12 spins arranged in a ring and an alternating chain 
Hamiltonian (solid lines). The exchange constants of J/ = 12.85 K and J^ = 4.35 K were 
determined by varying the exchange constants to minimize the square of the distance 
between the theoretical and experimental curves. The theoretical curve consisted of 3000 
points. A 250 point adjacent averaging procedure was applied to the first derivative of the 
theoretical curve (inset). A value of 2 was assumed for the Lande g factor. 



92 



4.4 High Field Magnetization Measurements 
The high-field, H <30 T, magnetization, M, of a BPCB powder sample (m = 
208.2 mg) normalized to its saturation value, Ms, is shown as a function of field and 
temperature in Fig. 4.20. Since the saturation magnetization was reached on our studies, 
we were able to measure and subtract a small, temperature-independent contribution 
(Xdiam ~ -2.84 X 10"* emu/mol), which is the same value obtained in the low field work 
(Section 4.2), by performing a linear fit to the data above 20 T. The data were acquired 
while ramping the field in both directions, and no hysteresis was observed. Although 
approximately 3000 points were acquired at each temperature, the data traces are limited 
to 150 points for clarity. The lines are fits using the 12 spin exact diagonalization and an 
alternating chain Hamiltonian, Eq. 4.2. The best fit exchange constants, which are listed in 
Table 4.4, have a systematic temperature dependence with J| increasing and Jj decreasing 
with increasing temperature. The theoretical curves adequately reproduce the 
magnetization data at the two highest temperatures of 3.31 K and 4.47 K. However, at 
the temperature of 1.75 K, the exact diagonalization curve deviates significantly fi-om the 
data at Hci =6.6 T and Hc2 =14.6 T. Data were also taken at 0.7 K; however, the exact 
diagonalization technique fails to produce a reasonable curve for the reasons discussed in 
Chapter 3, and consequently, that theoretical curve is not shown. It should be noted that 
the exchange constants, Ji ~ 13 K and J2 ~ 7.0 K, do not match the exchange constants 
obtained fi-om the susceptibility data, Ji ~ 13.7 K and J2 ~ 5.5 K. Similar results are 
obtained for magnetization measurements of single crystal samples. The high field, H < 
30 T, magnetization of a single crystal sample (m = 18.9 mg) with H \\ a-axis is shown in 



93 




Figure 4.20: The magnetization, M, of a BPCB powder sample (m = 208.2 mg) 
normalized to its saturation value, is shown as a function of field and temperature. A 
small diamagnetic correction has been subtracted fi-om the data. Although approximately 
3000 points were acquired at each temperature, the data traces are limited to 150 points 
for clarity. The lines are fits using the 12 spin exact diagonalization procedure and an 
alternating chain Hamiltonian. The exchange constants are listed in Table 4.4. Data were 
also taken at 0.7 K; however, the exact diagonalization technique fails to produce a 
reasonable curve for the reasons discussed in Chapter 3, and consequently, that theoretical 
curve is not shown. 



94 



CO 



1.0 


1 1 1 1 1 1 1 1 ■ 1 1 1 1 
' BPCB Single Crystal 


' ^^^^L^ 


^^^jS,^^^^9^W93U:3^ 


0.8 


■ H 


a-axis ^-^ 
1.55 K 


f\^V 4.33 K 




■ 


/ 


J 2.67 K 


0.6 


- 


F 


- 


0.4 


- 


I 


- 




■ 


M 


Experimental Data 


0.2 


- 


J^o° 




tzxaci uiaQonaiizaiion' 




^ 


^^ 


(Alternating 


0.0 


1 1 1 


1 1 1 1 1 1 1 1 1 1 


Chain Hamiltonian) 

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 







10 



15 

H(T) 



20 



25 



30 



Figure 4.21: The magnetization, M, of a BPCB single crystal sample (w = 18.9 mg) 
normalized to its saturation value, is shown as a function of field and temperature. A 
small diamagnetic correction has been subtracted fi"om the data. Although approximately 
3000 points were acquired at each temperature, the data traces are limited to 150 points 
for clarity. The lines are fits using the 12 spin exact diagonalization procedure and an 
alternating chain Hamiltonian. The exchange constants are Usted in Table 4.4. 



95 



1.0 - 



0.8 



_w 0.6 



0.4 



0.2 



0.0 




Experimental Data 
Exact Diagonalization Fit 
(Ladder Hamiltonian) 



10 



15 
H(T) 



20 



25 



30 



Figure 4.22: The magnetization, M, of a BPCB powder {m = 208.2 mg) normalized to its 
saturation value, is shown as a ftinction of field and temperature. A small diamagnetic 
correction has been subtracted irom the data. Although approximately 3000 points were 
acquired at each temperature, the data traces are limited to 150 points for clarity. The 
lines are fits using the 12 spin exact diagonalization procedure and a ladder Hamiltonian. 
The exchange constants are listed in Table 4.4. 



% 



0.8 



jp 0.6 



—r— I— r— r-F— r— I— T— p-i— T-r 

BPCB Single Crystal 



' 1 ' ' ' ' I 



H II a-axis 




_l I I I I I L 



10 



Experimental Data 
Exact Diagonalization 
(Ladder Hamiltonian) 



15 
H(T) 



20 



25 



30 



Figure 4.23: The magnetization, M, of a BPCB single crystal {m = 18.9 mg) normalized 
to its saturation value, is shown as a function of field and temperature. A small 
diamagnetic correction has been subtracted fi-om the data. Although approximately 3000 
points were acquired at each temperature, the data traces are limited to 150 points for 
clarity. The lines are fits using the 12 spin exact diagonalization procedure and a ladder 
Hamiltonian. The exchange constants are Usted in Table 4.4. 



97 



Table 4.4: The alternating chain parameters, J\ and J2, and ladder parameters, Ji and J\\, 
determined from fitting the magnetization vs. field data for a BPCB powder sample 
(w = 208.2 mg) and a single crystal sample (m = 18.9 mg) using the 12 spin exact 
diagonalization procedure. All exchange parameters listed are specified with an error of 
± 0.05 K, which is the iteration of the fitting routine. The chi^ value is the sum of the 
square of the difference of the theoretical and experimental curve for 3000 points where 
the saturation magnetization is normalized to one. 







r(K) 


/i(K) 


^2(K) 


chi^ 




d 

e 

Hi 


1 



a. 


0.7 


12.00 


7.55 


0.241 


1.75 


12.75 


6.85 


0.130 


3.31 


13.20 


6.70 


0.00773 


4.47 


13.50 


6.50 


0.00524 


1 
U 

t 


1.55 


12.00 


7.40 


0.1721 


2.67 


12.50 


6.70 


0.0320 


4.33 


12.35 


8.05 


0.0533 





1 



T(K) 


J,(K) 


J\\ (K) 


chi' 


0.7 


12.85 


3.65 


0.132 


1.75 


12.70 


4.00 


0.0132 


3.31 


12.55 


4.35 


0.00858 


4.47 


12.75 


3.95 


0.00289 


3 
t 

u 

1 

00 


1.55 


12.50 


3.85 


0.179 


2.67 


12.20 


3.75 


0.0744 


4.33 


12.70 


4.20 


0.0632 



98 



Fig. 4.21. Again, for the lowest temperature of 1.55 K, the theoretical curve deviates 
from the data at Hci and Hci- Because the signal size is approximately an order of 
magnitude larger at Hc2 than at Hci, the signal to noise ratio is also higher. Accordingly, 
the theoretical curves match the data at Hc2 better than at Hci. For BPCB powder, the 
signal size was not a limitation due to the relatively large sample mass. 

We can also fit the powder and single crystal magnetization data using the ladder 
Hamiltonian, Eq. 4.1. Figures 4.22 and 4.23 show the high field BPCB magnetization 
data from Figs. 4.20 and 4.21, except that the solid lines are fits using the 12 spin exact 
diagonalization procedure and a ladder Hamihonian. The exchange constants, which vary 
slightly with temperature, when compared to the alternating chain results, are listed in 
Table 4.4. For the powder data, the exact diagonalization procedure produced a 
reasonable result at the lowest temperature of 0.7 K because the ladder Hamiltonian is 
quasi-two dimensional, i.e. the correlation length is shorter than for an alternating chain. 
Although the ladder Hamiltonian works at lower temperatures, this fact alone is not 
indicative of the appropriateness of the model. However, the theoretical curves match the 
data better and the best fit exchange constants, Jx ~ 12.7 K and ^| ~ 3.9 K, are reasonably 
close to the values obtained from the susceptibility data, Ji ~ 13.2 K and J\\ ~ 3.9 K. In 
addition, the systematic change in the best fit parameters, as temperature is changed, is 
less when using the ladder Hamiltonian. In conclusion, the exact diagonalization fits to the 
magnetization data appear more self-consistent when using the ladder Hamiltonian 
suggesting that the ladder model is a more appropriate description of the magnetic 
exchange in BPCB. Although at this point we cannot determine if the ladder model is 



99 



correct, we can rule out the alternating chain description. Therefore, for the remaining 
portion of this chapter, only the ladder model will be discussed. 

The magnetization vs. field for BPCB powder at the lowest temperature of 0.7 K 
is shown in Fig. 4.24. At this temperature, it is possible to measure the critical fields, Ha 
= 6.6 ± 0.1 T and Hc2 =14.6 + 0.1 T, directly firom the graph by extrapolating fi-om the 
nearly linear dependence on field on either side of the critical field. The solid line in 
Fig. 4.24 represents the curve produced using the previously mentioned fit with Ji = 
12.85 K and J| I = 3.65 K, and the inset shows the first derivative of the magnetization 
d(M/Ms)/dH\s. field. It is clear that there is a feature at Ms/2, whose first derivative has a 
double humped structure with a local minimum in the first derivative at exactly Ms/2. This 
feature, as well as the entire magnetization curve, is symmetric around the field 10.6 T. 
From the inset of Fig. 4.24, the agreement between the predicted curve and the 
experimental data is evident. A 200 point adjacent averaging procedure was applied to the 
predicted derivative curve (inset) to smooth out additional spurious features that are a 
consequence of the limitations of the exact diagonalization technique (see Chapter 3). 

The feature at Ms/2 resembles the formation of a plateau. Cabra et al. [73] have 
stated the condition for the formation of plateaus in ladder systems as 

{S-(ms))=n , (4.8) 

where « is an integer, 5" is the total spin, and ms is the quantum spin number along each 
rung. According to Eq. 4.8, for a two-leg ladder, plateaus can exist only at a 



100 



1.0 
0.8 
_w0.6 
> 0.4 
0.2 
0.0 



— 1 — I — I — I — I — I — I — I — I — I — I — I — I — r 

Experimental Data 
Exact Diagonalization 
J =12.85 + 0.05 K 
J= 3.65 + 0.05 K 



-i — I — r 



rnmssBBStn 



" I , II mmnmrr 








J I I I I l_ 



5 10 15 20 

H(T) 

J I I I I I I I I I I I L. 



10 



15 

H(T) 



20 



25 



30 



Figure 4.24: The magnetization, M, of a BPCB powder sample (m = 208.2 mg) 
normalized to its saturation value, is shown as a flinction of field and temperature. A 
small diamagnetic correction has been subtracted irom the data. Although approximately 
3000 points were acquired at each temperature, the data traces are limited to 150 points 
for clarity. The solid line is a fit using the 12 spin exact diagonalization procedure and a 
ladder Hamiltonian with Ji = 12.85 ± 0.05 K and ^| = 3.65 ± 0.05 K. The agreement 
between the derivative of the predicted magnetization using the exact diagonalization 
method and the data (inset) is impressive. A 200 point adjacent averaging was applied to 
the predicted derivative curve to smooth out anomalous features that are a result of a 
limitation of the technique (see Chapter 3). 



101 



magnetization of and Ms. However, ladder systems can exhibit plateaus in their 
magnetization curves at rational fractions of the saturation magnetization due to 
frustration [73,75] or inter ladder coupling f [76]. In BPCB, the distance and the lack of 
superexchange mechanisms between ladders suggest that interladder coupling is not 
important at this temperature. Nevertheless, for a two-leg ladder, a diagonal fiaistration 
will produce a plateau at Ms/2 [77-78]. As stated earlier, because one diagonal distance is 
much shorter, we will consider only one frustration exchange Jf- Mila [77] has calculated 
the properties of a two-leg frxistrated ladder using the results of Totsuka [79] and 
Tonegawa [80]. The values of the critical fields and the width of the plateau at Ms/2 are 
given as [77] 



and 



^C\ = J_L -Ji\ + Jf ^2 , 



■"C2 — -^i + ^J\\ 5 



AH = 4;r • Jp exp 



TT 



2y[i 



**J II ~~ ^*J p 



,1/2 



(4.9) 
(4.10) 



(4.11) 



By estimating the value of AH ~ 4 T using Fig. 4.24, Eqs. 4.9 through 4.11, having only 
three unknowns, can then be solved. It is clear, from Eqs. 4.9 and 4.10, that the value of 
Jf cannot be very large (Jf « J||) to maintain the symmetry observed in the magnetization 
curves. The theory dictates that, in order to produce a plateau, the fioistration exchange 



102 



must be larger than the leg exchange, Jf > 3Ji|/2. Solving for the exchange constants, we 
obtain: Ji = 11-58 K, ^j = 4.42 K, and Jf = 4.50 K. Figure 4.25 shows the molar 
magnetic susceptibility of BPCB powder with the solid line calculated using the previously 
mentioned parameters. Clearly, frustration cannot consistently explain the susceptibility 
vs. temperature and magnetization vs. field results simultaneously. From similar 
arguments, we can estimate an upper bound for the fiiastration exchange of Jf < 0.5 K. 



10 



25 


1 


. 1 1 1 1 1 1 1 1 1 1 _ 
O Experimental Data 




■ /^ 






In Lxact uiagonaiization uurve using.. 


20 


" / 


&. J^ = 1 1 .58 K : 




■ H 


^fe^ J|i= 4.42 K 


O 

E 15 

"B 
E 

<? 10 

o 

?^ 
5 


- ) 

- ) 

'■■I 

-<) 

-) 
() 


^^ Jp = 4.50 K 

BPCB Powder ~ 


n 


:8 

1 1 


H = 0.1 Tesia - 
■ 1 1 . " 



20 



30 40 

T(K) 



50 



60 



70 



Figure 4.25: The BPCB powder molar magnetic susceptibility vs. temperature (m = 166.7 
mg) in an applied field of 0.1 T. The solid line is a 12 exact diagonalization curve 
produced with the frustrated ladder parameters of Jj_ = 11.58 K, J\i = 4.42 K, and 
Jf = 4.50 K. The susceptibility and magnetization results cannot be consistently explained 
by including frustration in a ladder or alternating chain model. 



103 



It is possible, by treating J\\IJl as a perturbation, to map the low energy states of 
the ladder Hamiltonian onto the exactly solvable 5=1/2 XXZ chain [79]. This technique, 
which was described in Chapter 3, was used to fit the M{H,T) data of Cu(Hp)Cl [14]. It 
should be noted, however, that the magnetization results for Cu(Hp)Cl were not 
symmetric around Ms/2 and the derivative of the magnetization data was not provided in 
the relevant paper [14]. Fitting the BPCB magnetization data using this method is 
particularly attractive since the mapping is symmetric around Ms/2 and therefore all 
quantities calculated fi-om the XXZ Hamiltonian will also be symmetric around the center 
field. This symmetry of the magnetization curves for a proposed ladder material has never 
before been reported and is evidence of the symmetry of the magnetic exchange. The 
normalized magnetization, MIMs, vs. field for BPCB powder and single crystal are shown 
as a function of temperature in Figs. 4.26 and 4.27, respectively. The solid lines were 
obtained by numerical integration of the Bethe ansatz equations [67] and the exchange 
parameters were then used to calculate x(7) (see Figs. 4.28 and 4.29) for comparison with 
the susceptibility data. All of the data could be reasonably reproduced using a single set of 
exchange constants, namely Ji = 13.3 ± 0.2 K and ^j = 3.8 + 0.1 K. To leading order 
[77,81], 

gHBHcilkB = J^-J\\, (4.12) 

and 

gUBHcilks^JL^Uw. (4.13) 



104 



JP 0.6 - 




Figure 4.26: The magnetization, M, of a BPCB powder {m = 208.2 mg) normalized to its 
saturation value, is shown as a function of field and temperature. A small diamagnetic 
correction has been subtracted from the data. Although approximately 3000 points were 
acquired at each temperature, the data traces are limited to 150 points for clarity. The 
lines are predictions of an effective XXZ chain when Jj, = 13.3 K and J\\ = 3.8 K. 



105 



1.0 


BPCB Single Crystal 


■■■'!■■■ 'I ■ ■ ' ' 


^^^^^^^^-S>.^fia£^ 


0.8 


H a-axis -"^^ 
1.55 K M' 


J\ 4.33 K - 
^ 2.67 K 


0.6 


°//3b 


- 


0.4 


M 


- 




,W^ c 


Experimental Data 

YY7 Oh^in 


0.2 


■* iQS^^/Si 


r\KL. unain 






J^=13.3K 


0.0 


> 


J|i= 3.8 K 
1 1 1 1 1 1 1 1 1 1 1 1 1 1 



10 



15 
H(T) 



20 



25 



30 



Figure 4.27: The magnetization, M, of BPCB single crystal (w = 18.9 mg) normalized to 
its saturation value, is shown as a function of field and temperature. A smaU diamagnetic 
correction has been subtracted from the data. Although approximately 3000 points were 
acquired at each temperature, the data traces are limited to 150 points for clarity. The 
lines are predictions of an effective XXZ chain when Ji = 13.3 K and J\\ = 3.8 K. 



106 



25 



20 
O 

E 

(D 



10 



T 



T 



T 



T 



T 



T 



Experimental Data 

Exact Diagonalization Curve using: 

J^ = 13.3 K 

J|i= 3.8 K 

Impurity Cone. = 0.7 % 




BPCB Powder 
g AV = 0.1T 



J I L 



X 



_i I 1 L 



_L 



_L 



10 20 30 40 50 60 70 80 90 100 

T(K) 

Figure 4.28: The molar magnetic susceptibility vs. temperature for BPCB powder (w = 
166.7 mg). The sample was zero field cooled to 2 K and then measured in a field of 0. 1 T. 
A small constant diamagnetic contribution of Xdmm = -2.84 x 10"^ emu/mol has been 
subtracted. The solid line represents the best fit by comparing the experimental curve with 
curves produced using the 12 spin exact diagonalization method discussed in the text with 
parameters A = 13.3 ±0.1 K and ^ =3.8 ±0.1 K. 



107 



25 



20 



15 



O 

E 
E 

CD 

CO 

b 10 



5 - 



O 



I ' I ' I ' 

Experimental Data 
Exact Diagonalization Curve using: 
J^=13.3K 
J|i= 3.8 K 
Impurity Cone. = 0.7 % 




BPCB Single Crystal 
H = 0.1T||a-axis 



20 



40 60 

T(K) 



80 



100 



Figure 4.29: The molar magnetic susceptibility vs. temperature for a BPCB single crystal 
(m = 46.9 mg). The sample was zero field cooled to 2 K and then measured in a field of 
0.1 T. A small constant diamagnetic contribution of /diam = -3.16 x 10^ emu/mol has 
been subtracted. The solid line is a curve produced using the 12 spin exact diagonalization 
method discussed in the text with parameters Ji = 13.3 ± 0.1 Kand^i = 3.8 ± 0.1 K. 



108 



Using the previously mentioned parameters, we obtain Hci = 6.6 T and Hci = 14.6 T, 
which are identical with the experimental results. The agreement between the theory and 
the data is impressive, considering that a single set of exchange constants can adequately 
explain all of our data. In conclusion, BPCB is a strongly coupled ladder system with Jj_ = 
13.3 ± 0.2 K, Jii = 3.8 ± 0.1 K, and Jf < 0.5 K. 

Figure 4.30 shows the &st derivative of the magnetization, d{M/Ms)/dH, of BPCB 
powder {m = 208.2 mg) for the lowest temperature of 0.7 K normalized to its saturation 
value, as a function of field. The number of points has been limited to 150 for clarity. The 
solid line is the prediction for an effective XXZ chain when Jj. = 13.3 K and J\\ = 3.8 K. 
This symmetric double bump structure and its evolution with temperature have been 
studied theoretically [18] but have not been observed previously in 5 = 1/2 two-leg ladder 
materials. Even though our theoretical curve somewhat overestimates the sharpness of 
d{M/Ms)/dH, the overall agreement is excellent, and involves no adjustable parameters 
once Hci is defined. Furthermore, the fact that we see only one feature at Ms/2 between 
Hci and Hc2 is evidence that our strongly interacting dimers are not coupling to form 2D 
[82] or 3D [83] networks. The first derivative of the normalized magnetization, 
d(M/Ms)/dH, of a BPCB single crystal (m = 18.9 mg), with H || a-axis for the temperature 
of 1.55 K, is shown in Fig. 4.31. Again, the solid line represents the prediction of the 
XXZ chain using the same parameters as stated earlier. At this temperature, the first 
derivative is just beginning to develop a minimum at Ms/2. Overall, the first derivative 
data agree with the calculated curves and the slight discrepancy may be due to uncertainty 



109 



in the temperature measurement or the low energy state approximation used to map the 
ladder Hamiltonian onto an effective XXZ chain. 

As discussed in Chapter 3, the number of spins used in the exact diagonalization 
procedure should be as large as possible so that the correlation length between spins does 
not exceed the total length of the model system. However, the number of spins used in 
the calculation is limited by the computing power available. At low temperatures, the 
correlation length can become arbitrarily large and the accuracy of the calculation 
decreases accordingly. It is important to know how many spins are necessary, using the 
ladder Hamiltonian, to accurately model BPCB. Considering that we have accurate values 
for the exchange parameters using the XXZ solution, we can compare the results using the 
two methods. In Fig. 4.32, the parameter ..^i, determined by fitting the 1.75 K powder 
magnetization data using the exact diagonalization procedure, is plotted as a function of 
the number of spins used in the calculation. The solid line indicates the value of J\\ 
determined by mapping the ladder Hamiltonian onto the XXZ chain. The J\\ value 
overshoots and then appears to approach a constant value slightly above J|| = 3.8 K as the 
number of spins is increased. However, it is impossible to determine if both methods will 
produce the same value without using a greater number of spins {N ~ 20). The ladder 
exchange parameters determined fi-om fits using the exact diagonalization technique are 
systematically lower for the rung exchange Jx(") < Ji(«-*oo) and systematically higher for 
the leg exchange J\\{n)> J\\{n^<x)). 



no 



0.20 



0.15 



X 0.10 



-D 0.05 



0.00 



T — I — 1 — I — I — I — I — I — I — I — 1 — I — I — I — I — I — I — I — r 
X Experimental Data 

Prediction from XXZ Model using 

J, = 13.3 K and >;= 3.8 K 




Figure 4.30: The first derivative of the magnetization, d{M/Ms)/dH, of BPCB powder 
(m = 208.2 mg) at a temperature of 0.7 K normalized to its saturation value, is shown as a 
function of field. The number of points has been limited to 150 for clarity. The solid line 
is the prediction for an effective XXZ chain when A = 13.3 K and ^| = 3.8 K. 



Ill 



0.15 



0.10 



X 
■a 



0.05 - 



0.00 



— I 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 r 

X Experimental Data 

Prediction from XXZ Model using 

J^ = 13.3 K and J|| = 3.8 K 

^ .... A 




Figiire 4.31: The first derivative of the magnetization, d{M/Ms)/dH, of a BPCB single 
crystal {m = 18.9 mg) normalized to its saturation value, is shown as a function of field. 
The number of points has been limited to 150 for clarity. The solid line is the prediction 
for an effective XXZ chain when Ji = 1 3.3 K and Jn = 3.8 K. 



112 



4.2 


- 


1 


1 

• 


1 


' 1 ' 1 

• 


4.0 


- 








• 


3.8 








• 














3.6 


- 








- 


3.4 


- 








- 


3.2 


- 








BPCB Powder 
T=1.75K 


3.0 


- 


• 

1 


1 


1 


Exact Diagonalization 

1 , 1 



6 8 10 

Number of Spins 



12 



Figure 4.32: The parameter J\\ determined by fitting the T = 1.75 powder magnetization 
data using the exact diagonalization procedure vs. the number of spins used in the 
calculation. The solid line indicates the value determined using by mapping the ladder 
Hamiltonian onto the XXZ chain. The J\\ value appears to approach a constant value that 
is slightly higher than 3.8 K as the number of spins is increased. However, it is impossible 
to determine if both methods will produce the same exchange constants using only 12 
spins in the exact diagonalization. 



113 



4.5 Universal Scaling 
At Hch BPCB undergoes a transition from gapped dimer pairs to a gapless 
Luttinger liquid phase with fermionic excitations, where the magnetization is proportional 
to the fermion density [68,84,85]. The transition can be described as a condensation of a 
dilute gas of bosons (dimers), where quasiparticle interactions are irrelevant at the 
transition point. At Hc2, an analogous situation exists where the transition is between the 
Luttinger liquid and spin polarized phases. Near Hci and Hc2, the ID magnetization is 
predicted to obey the universal scaling law (assuming JJJ\\ » 1) 



M = g^gS 



B'-'Z 






kj _ 



^ 



(4.14) 



where d= \ is the dimensionality of the system, m is the quasiparticle mass, and ^ is the 
fermion density [68]. The fermion density can be calculated using the definite integral, 

^(^) = -|^>'^r^' (4.15) 

which must be solved numerically with / = gfj.B{H-Hci)lkB as the independent variable. 
However, the quasiparticle mass is unknown and must be estimated using a dispersion 
relationship. Reigrotzki et al. [81] have written the dispersion relationship for an 5 = 1/2 
ladder system as 



114 



£ {k) I kg =Ji^+ Jh cos(ka) + 



•^ II 



v-^^y 



{3-cos(2ka))+--- . (4.16) 



By comparing this dispersion relationship, expanded around the minimum in the dispersion 
curve, ka = 7t, {a ]s the distance between rungs) with the quasiparticle dispersion 
relationship given by Sachdev et al. [68], namely 



£{k) = ^ + ~ gMsSzH , 

2m 



(4.17) 



we can solve for the quasiparticle mass and spin gap A. To second order in J\\, we obtain 
the result 



fh^e^ 



\ '" J 



~ '^B^W 



1 + 



8 J^ 



{ka-n:y , 



(4.18) 



and 



A/ kg =J^ - J|| +- 



•^ II 



yJ.j 



(4.19) 



Now, it is trivial to plug the result from Eq. 4.18 into Eq. 4.14 to obtain the scaling 
relationships at Hci and Hci, i.e. 



115 



M{H,T)/Ms=pT/J,^[gMsiH-Hc,)/kj] , (4.20) 

and 

{M{H,T)/Ms-'i)=pT/J^^^[gMs{H-Hc2)/ksT] . (4.21) 

This universal scaling, which is valid over the field region g^alH - Hc\/T < kaJw/T and the 
temperature region T <J\\, is compared to the data in Fig. 4.33, where the agreement is 
impressive. It is noteworthy that the scaling shown in Fig. 4.33 has been theoretically 
predicted [68], and is not the result of extracting scaling variables on the basis of the data. 
For the material Cu(Hp)Cl [14], different scaling relationships were used to describe the 
data at the fields Hci and Hc2 and neither scaling relationship was derived fi-om theoretical 
expectations. For BPCB, a deviation fi-om scaling is observed at 0.7 K, which suggests 
that other weak interactions may be present. The data at 4.47 K, while slightly above the 
valid temperature range, appear to obey the scaling theory. An analogous situation exists 
for the magnetization data shown in Fig. 4.34 for single crystal BPCB with H parallel to 
the a-axis scaled in the vicinity oiHci and Hc2- Here the 4.33 K data appear to match the 
theoretical scaling curve, while some deviation is apparent at the lowest temperature of 
1.55 K. It is also interesting to note that the data at Hci and Hc2 in Fig. 4.34 are not as 
symmetric around \H - Hc\ as the data shown in Fig. 4.33. As mentioned previously, 
because the mass is much smaller for the single crystal sample, the magnetization 
measurement is more accurate for the single crystal sample at the higher field oi Ha than 
at the field Hct. Figure 4.35 shows the magnetization at Hci vs. temperature for BPCB 



116 




g^3(H - H,,) / k^T 



Figure 4.33: The magnetization data scaled in the vicinity of Hci (lower curve) and Hc2 
(upper curve) for BPCB powder. The solid lines are the prediction of Sachdev et al. [68] 
using the critical fields of Hci = 6.6 T and Hc2 = 14.6 T. 



117 



CM 



1.0 



- 0.8 " 



- 0.6 m 



- 0.4 !^ 



0-2 ^S 



0.0 
-1 1 

Figure 4.34: The magnetization data scaled in the vicinity of Hci (lower curve) and Hc2 
(upper curve) for single crystal BPCB with H parallel to the a-axis. The solid lines are the 
prediction of Sachdev et al. [68] using the critical fields of i/c/ == 6.6 T and Hc2 = 14.6 T. 



w 




118 



0.30 




T(K) 



Figure 4.35: The magnetization at Hci vs. temperature for BPCB powder and single 
crystal indicating the T"' scaling behavior. The solid line is the prediction of Sachdev et 
al. [68] using the critical field oiHci = 6.6 T. 



119 



powder and single crystal indicating the T"'^ scaling behavior predicted for an .S" = 1/2 
ladder material [76]. The solid line is the prediction of Sachdev et al. [68] using the 
critical field oiHci = 6.6 T. Again, the agreement is admirable, providing further evidence 
that BPCB is a two-leg spin ladder with a negligible interladder coupling as low in 
temperature as 0.7 K, i.e. J' «J\\. 

4.6 Neutron Scattering 
Neutron diflfraction measurements on the material Cu(Hp)Cl were previously able 
to confirm the presence of a magnetic spin gap as well as elucidate details of the magnetic 
exchange [15,17]. Similar experiments on BPCB have been performed at the High Flux 
Isotope Reactor HB-3 beamline at Oak Ridge Nation Laboratory. The sample was a 
single crystal of deuterated BPCB with a mass of approximately 1 . 1 g. The x-ray crystal 
structure fi-om the paper by Patyal et al. [20] was used as the initial structure in the 
neutron studies. The crystal structure was verified and no structural transitions occurred 
down to the lowest temperature of 1.6 K. Two types of neutron scattering experiments 
were performed. First, inelastic neutron scattering [72] scans were conducted with a fixed 

incident energy and fixed Q along the ladder or rung direction of BPCB at a temperature 

below the spin gap, {T « A/^b). Second, at temperatures above and below the gap, 
quasi-elastic neutron scattering scans [86] were performed as a fimction of wavevector, 
Q , with fixed incident energy and integrated final energy. 



120 



Figure 4.36 shows neutron counts versus change in energy, AE = Ei - Ef, with a 
fixed incident energy of 14.7 meV along the ladder direction, Q = [2 0]. The solid line 
is a Gaussian fit to the data indicating a resolution of 1 .04 meV, where the non-zero center 
of the Gaussian fit is produced by a small misalignment of the analyzer crystal. The peak 
at AE = is the elastic (Bragg) peak. If the spin gap. A, is produced by magnetic 
interactions along the ladder direction, we would expect an inelastic neutron peak due to 
magnetic scattering at AE/ke = Mka = 9.5 K = 0.82 meV in Fig. 4.36. However, inelastic 
scattering scans along the ladder direction ^=[200] and approximate rung direction Q 
= [0 2] produced a result that was indistinguishable fi-om the background. Apparently, 
the resolution was insufScient to identify the relatively small magnetic inelastic peak within 
the elastic scattering peak. 

Figure 4.37 shows neutron counts versus change in wavevector, Q = [10 l\, with 
a fixed incident energy of 30.5 meV and integrated final energy at a temperature of 1.6 K. 
The Bragg peaks are evident as sharp peaks at integer wavevectors produced by the 
periodicity of 5 = 1/2 Cu^"^ spins. By compiling 92 scans at this same temperature, the 
grayscale contour map in the a-c plane of BPCB was produced. Fig. 4.38. The grayscale 
mapping assigns the highest number of counts as white and zero counts as black. The 
region at the lower right corresponds to wavevectors that are beyond the angular limits of 
the spectrometer. The broad circular band is an ~ 1% concentration impurity powder 
dififraction line with a lattice constant of approximately 12.6 A, most likely produced by 
fi-ee piperidinium bromide. Magnetic scattering should be evident as a sharp peak, not 



121 



located at an integer wavevector, that disappears when the temperature is raised above the 
gap temperature T > A/yts. In addition, quasi-elastic magnetic scattering should be located 
along the ladder direction and near the wavevector Q = [200]. 

Figure 4.39 is a contour map of neutron scattering counts in the a-c plane of single 
crystal BPCB versus wavevector at a temperature of 45 K, above the gap temperature. 
By subtracting the number of counts in Fig. 4.39 from the number of counts in Fig. 4.38 at 
each wavevector, Fig. 4.40 was produced. The Bragg peaks and powder diffraction ring 
are still evident after subtraction due to the temperature dependent scattering or Debye- 
Waller effect. Unfortunately, magnetic scattering is not evident in the subtracted data. In 
the fiiture, waiting a longer time at each wavevector and using a larger single crystal 
sample or array of single crystal samples should improve the experimental sensitivity. 



122 



4000 



CO 

■♦-« 

c 

O 

O 

c 

2 

*^ 

3 
<D 



3000 



2000 



1000 



—I ' \ ' r 

O Experimental Data 

Gaussian Fit 

1 .04 meV width 



1 1 ' 1 

BPCB Single Crystal 
£ = 14.7meV 



Q = [2 0] 




-1 1 

AE = E,-E^{mey) 



Figure 4.36: Neutron counts versus change in energy, AE, with a fixed incident energy of 
14.7 meV along the ladder direction, Q = [2 0]. If the spin gap observed in the 
susceptibility measurements was produced by magnetic ordering along the ladder 
direction, an inelastic neutron peak should be evident at approximately 1 meV. 



123 



o 



25 



20 



(A 

^-» 

C 
O 

O 

c 
o 

0) 



. BPCB Single Crystal 
7=1.6K 
E, = 30.5 meV 
Integrated Final Energy 



Q 



« 15 



10 



1.0 



1.5 




J L. 



2.0 

QA20f\ 



2.5 



3.0 



Figure 4.37: Neutron counts versus change in wavevector, Q = [2 I], with a fixed 
incident energy of 30.5 meV and integrated final energy. The Bragg peaks are evident as 
sharp resonances at integer wavevectors. 



124 



o 
o 



q _ 






T=1.6K 






o 


1 A /"*% A A 


^. 


^'"^V^^ 


- 


6 g ^ pv 









A A ^— ^ >H^ / A. / A y ' 


^ <^ A 




2- 




^ 


^ 1 

o < 


- 


?""(] *5(y^»^r^ 


► "^^^ 


^^ 


1 _ 


6 ^ 11 


m^^^ 


i^-~ 


^^ 


1 


r '^ 








A 




>Xa n?^ pB 


^^^^^^1 




0- 


.. I 


1 :x )fv^M 




Iplp^ 







[0 01] 



Figure 4.38: Neutron scattering counts in the a-c plane of single crystal BPCB versus 
wavevector at 1.6 K. The incident energy was 30.5 meV and the final energy was 
integrated. The grayscale mapping assigns the highest number of counts as white and zero 
counts as black. The region in the lower right corresponds to wavevectors that are 
beyond the angular limits of the spectrometer. The white peaks on integer lattice spacings 
are Bragg peaks. The broad circular band is produced by powder diffraction (see text). 



125 



o 
o 




Figure 4.39: Neutron scattering counts in the a-c plane of single crystal BPCB versus 
wavevector at 45 K. The grayscale mapping assigns the highest number of counts as 
white and zero counts as black. The region in the lower right corresponds to wavevectors 
that are beyond the angular limits of the spectrometer. 



126 



o 
o 

sz 



^lo 



(T = 1 .6 K data) - (7 = 45 K data) 




no o U k377A vTr^^^r^Av 



o ;^o7:ooV 




o 



0-^ 




./ ▼ 





^ 



.0 « 




0"O 0. 



'o %r) o/^r. ^ ^%: '0 « 




Figure 4.40: The neutron scattering counts in the a-c plane of single crystal BPCB versus 
wavevector at 1.6 K minus the number of counts at 45 K. The grayscale mapping assigns 
the largest positive difiference as white, the largest negative difference as black, and zero 
as gray. The Bragg peaks and powder diffraction ring are still evident after subtraction 
due to the Debye- Waller effect. 



CHAPTER 5 
MAGNETIC STUDY OF A POSSIBLE ALTERNATING CHAIN MATERL^. 



Alternating chain materials consist of atoms that have a total magnetic spin 
arranged roughly in a linear geometry with the exchange between spins, akemating 
between the values J\ and J2 mediated by superexchange mechanisms and dipole-dipole 
interactions. Antiferromagnetic alternating chains that consist of fractional spin values at 
each site {S = 1/2, 3/2, ...) will have a spin gap, to first order A/fe = Ji - Ji, and at low 
temperatures, T « A/ke, dimer pairs of the spins will form 5 = singlet states. The 
macroscopic magnetic susceptibility of these materials wiU be nearly indistinguishable from 
that of other low dimensional gapped materials [71], and several experimental materials 
exist that approximate an alternating chain magnetic structure [87-90]. This chapter 
presents the magnetic properties of catena(dimethylammonium-bis(}i2-chloro)- 
chlorocuprate), (CH3)2NH2CuCl3, hereafter referred to as MCCL, whose crystal structure 
was first determined by Willett [26]. The room temperature crystal structure of MCCL is 
formed hy S= 1/2 Cu^^ spins arranged in a zig-zag pattern with the distance between spins 
oscillating between two values, suggesting that the magnetic exchange resembles an 
alternating chain. Consequently, magnetic studies were undertaken to determine the exact 
nature and magnitude of that exchange. This chapter is divided into four sections, with 
the synthesis and crystal structure of MCCL introduced in Section 5.1, including 
preliminary neutron diffraction structural data obtained at Oak Ridge National Laboratory 

127 



128 

(ORNL). Electron paramagnetic resonance (EPR) measurements performed by Dr. 
Talham's research group in the Department of Chemistry at the University of Florida are 
presented in Section 5.2. Low field susceptibility measurements and high field 
magnetization experiments are presented in Sections 5.3 and 5.4, respectively. 

5.1 Structure and Synthesis of MCCL 
The material MCCL is formed by the slow evaporation of a 1:1 solution of 
CuCl2»2H20 and (CH3)2NH2C1 in methanol over a period of approximately two weeks. 
Single crystal samples with a mass of 50 mg as well as several grams of powder were 
synthesized. In addition, deuterated single crystal and powder samples were prepared for 
neutron scattering experiments. Samples appear reddish brown in color and single crystal 
samples have a flat irregular shape that is roughly rectangular. MCCL has a molecular 
weight of 216.00 grams per mole and a density of 1.94 g/cm^. The crystal structure of 
MCCL has been determined by X-ray diffraction experiments [26] to be monoclinic at 
room temperature, T ~ 300 K, with approximately planar Cu2Cl6~ dimers forming chains 
along the a-axis. This structure, as viewed along the b-axis, is shown in Fig. 5.1 [26]. 
The structure repeats every two S = 1/2 Cu^^ spins, and the exchange between spins is 
mediated by two bridging chlorine atoms with the distance alternating between the values 
3.441 A and 3.547 A. The bridging chlorine atoms form an angle of approximately 90° 
between two Cu atoms suggesting an antiferromagnetic exchange [91]. Chains in the a-c 
plane are separated fi-om each other by dimethylammonium ions (CH3)2NH2 with an 
average separation distance of 7.25 A. Adjacent layers are stacked along the b-axis 
separated by Cl-Cl van der Waals contacts with 8.63 A between chains. The distance 



129 

between chains, as well as the lack of superexchange mechanisms, suggest that the chains 
are magnetically isolated. Preliminary neutron diffraction work has been performed on 
approximately 1 . 1 grams of deuterated d-MCCL powder verifying the room temperature 
crystal structure previously determined by Willet [26]. A three dimensional representation 
along the b-axis, using the room temperature crystal structure, is shown in Fig. 5.2 where 
the hydrogen positions have been calculated. The Cu atoms (dark spheres) form a chain 
along the a-axis running vertically in Fig. 5.2. Further neutron scattering experiments have 
shown that MCCL undergoes a structural transition from monoclinic, space group I2/a, to 
triclinic, space group PT, at approximately 250 K. However, the detailed crystal 
structure below 250 K has not been determined. In addition, a second structural transition 
may occur between 50 and 1 1 K. The structural transitions are most likely caused by 
freezing of the rotational and vibrational modes in the (CH3)2NH2 groups, thereby altering 
the distance between chains in the a-c plane. 

The crystal structure of MCCL suggests that the magnetic exchange can be 
described by an Heisenberg alternating chain Hamiltonian, 



Nil N/2-] 

- T 
chain 



= J^llK-^■^2.+J2Y}2rK.^+gMstfir^ , (5-1) 



/=1 /=1 ;=I 



with parameters, J\ and Jj. Because the chains are isolated from each other, the magnetic 
behavior of MCCL should agree well with an alternating chain model. The distance 
between Cu ^ spins is nearly a constant and consequently the exchange parameters, J\ and 



130 




Copper 
Clorine 



o 



Nitrogen 
Carbon 



Figure 5.1: The room temperature crystal structure of MCCL as viewed along the b-axis 
by Willet [26]. The magnetic S= \/2 Cu^^ ions form chains parallel to the a-axis. 



131 




o 



Cu 



CI 



A N 

O c 



O H 



Figure 5.2: A three dimensional view of the crystal structure of MCCL along the b-axis. 
The 5=1/2 Cu^^ spins are indicated by the dark spheres. The hydrogen positions are from 
theoretical predictions. 



132 

J2, should be roughly the same magnitude. The fits in this chapter were performed using 
the 12 spin exact diagonalization technique described in Chapter 3 and the above 
Hamiltonian, Eq. 5.1. 

5.2 Electron Paramagnetic Resonance 

A microscopic probe such as electron paramagnetic resonance is usefiil in 
determining the nature of the structural transitions observed in the neutron diffraction 
experiments. In addition, EPR measurements will provide a value for the Lande g factor 
which will simplify fitting the susceptibility data. To this end, EPR measurements were 
performed with d-MCCL powder and protenated single crystal samples. Figure 5.3 is a 
typical EPR trace showing the first derivative of intensity vs. field at a fi-equency of 
9.267 GHz and a temperature of 5 K, and with the field along the long growth axis of a 
MCCL single crystal (m = 18.6 mg). After integration, the EPR spectrum of MCCL 
consisted of a single symmetric broad line at all temperatures and orientations, see Fig. 
5.4. The long growth axis is most likely to be the crystallographic a-axis, but this 
conjecture remains to be verified. 

The integrated EPR line intensity (area under the curve in Fig. 5.4) as a fimction of 
temperature with the field along the long growth axis of a MCCL single crystal (m = 
18.6 mg) is shown in Fig. 5.5. The intensity is proportional to the number of electronic 
spins in the sample that are aligned with the magnetic field and hence is proportional to the 
susceptibility. In Section 5.3, the susceptibility of MCCL will be discussed in detail. 
Similar susceptibility curves are obtained for EPR measurements with the field 
perpendicular to the long growth axis as well as for powder samples (see Fig. 5.6). 



133 



2 - 



3 
(0 



1 - 



-1 - 



I 1 1 . 1 1 1 1 1 1 
A f= 9.267 GHz 


/ \ MCCL Single Crystal 


/ 1 H II Long Growth Axis 


/ 1 r=5K 


1 < 



1000 



2000 



3000 4000 

Field (G) 



5000 



6000 



Figure 5.3: The first derivative of EPR intensity vs. field at a fi-equency of 9.267 GHz and 
a temperature of 5 K with the field along the long growth axis of a MCCL single crystal 
(m = 18.6 mg). 



-5- 

>> 

(0 

c 

a: 

Q. 
LU 



ou 


1 . 1 • 1 1 1 > 1 1 

f= 9.267 GHz 


25 


/\ MCCL Single Crystal - 




/ \ H II Long Growth Axis ■ 


20 


/ \ r=5K 


15 


/ \ " 


10 


/ \ ' 


5 


y \ - 





1 . L ,1,1,1, 



1000 2000 3000 4000 

Field (G) 



5000 



6000 



Figure 5.4: The EPR intensity vs. field at a fi-equency of 9.267 GHz and a temperature of 
5 K with the field along the long growth axis of a MCCL single crystal (m = 18.6 mg). 



134 



70 



^. 60 
^ 50 - 
I 40 



30 - 



a: 

Q. 
LU 



■§ 20 
CO 

O) 10 
0) 

c 
— 



1 ' 1 ' 1 


1 1 ' 1 ' 1 1 1 


fc 


f= 9.267 GHz 


- o 


— 




MCCL Single Crystal 


o 


H II Long Growth Axis 


o 


- 





- 


o 


■ 


_ o 


_ 


o 




o 




cb 




"^o 


- 


: 


°°°°°°°°-oooo „„ • 


1,1,1 


1 



50 



100 



150 200 
T(K) 



250 



300 



Figure 5.5: The integrated EPR line intensity vs. temperature at a frequency of 
9.267 GHz with the field along the long growth axis of a MCCL single crystal 
(m= 18.6 mg). 



(0 



60 



50 - 



w 40 
0) 

-S 30 1- 

cc 

Q. 
LU 

T3 
0) 

2 10 h 

O) 
Qi 

^ oj- 



20 - 



- 


1 ' 


1 ' 1 


' 1 


' 


1 ' 1 ' 1 


_ 


- 


o 
o 








f= 9.272 GHz 


■ 


- 


9> 

o° 
o 








d-MCCL Powder 


- 


~ 


o 










— 


- 


o 










. 


- 


o 
o 










- 


- 


c 


o 








- 


- 




\. 








: 


- 







o°8o 


o 


o O 


. 


— 


1 


1 . 1 


I 1 


i 


o o 

1 1 1 . 1 


- 



50 



100 



150 

T(K) 



200 



250 



300 



Figure 5.6: The integrated EPR line intensity vs. temperature at a frequency of 
9.272 GHz for d-MCCL powder {m = 12.0 mg). 



135 

Figure 5.7 shows the Lande g factor, obtained by the central position of the EPR 
line, as a fUnction of temperature with the field parallel to the long growth axis of a MCCL 
single crystal (w = 3.2 mg). The structural transition identified in the neutron dififraction 
experiments is clearly observed at approximately 245 K as a sudden change in the Lande g 
factor. From 10 to 50 K, where a second structural transition possibly exists, the g factor 
appears nearly constant, g = 2.10 + 0.01. A summary of the value of g for both single 
crystal and powder samples is given in Table 5.1. For each sample, the values at 200 K 
are given since the Lande g factor is a fiinction of temperature above 250 K. 



Table 5.1: A summary of the mass, m, Lande g factor, linewidth, A//p-p, and structural 
transition temperature for each sample determined fi-om EPR measurements. The symbols 
"d" and "p" represent deuterated and protenated samples, respectively. 





Powder 


// II Long 
Growth Axis 


//±Long 
Growth Axis 


mass (mg) 


12.0 (d) 


3.2 (p) 


18.6 (p) 


g at 200 K 


2.08 ± 0.01 


2.10 ±0.01 


2.00 ±0.01 


width at 200 K (G) 


386 ±3 


492 ±3 


416±3 


structural transition 
temperature (K) 


240 ±5 


245 ±5 


255 ±5 



The EPR linewidth is determined fi-om the difference in the peak positions in the 
derivative of intensity as a fiinction of magnetic field, see Fig. 5.3. After plotting the EPR 
linewidth as a fiinction of temperature with the field parallel to the long growth axis of a 
MCCL single crystal {m = 3.2 mg), we obtain Fig. 5.8. Again, the structural transition at 



136 



2.2 



2.1 O 



O 

0} 
T3 

TO 2.0 



— 1 1 > 1 1 1 1 1 ' 1 


_ 


eS— °°°«^°°°°°°°°Oooooo 


_ 


o 




o 




o 




f= 9.267 GHz ° 





- p-MCCL Single Crystal 


■ 


H II Long Growth Axis 

1,1,1,1,1 


1 



1.9 

50 100 150 200 250 300 

T(K) 

Figure 5.7: The Lande g factor vs. temperature determined from the EPR line position at 
a frequency of 9.267 GHz with the field parallel to the long growth axis of a p-MCCL 
single crystal (w = 3.2 mg). 



O 



a. 

Q. 



700 


1 1 1 1 1 
o 
o 


T 








I 




1 


• 1 


600 


o 
' o 












o 


oo 


- 


500 


~ o 










o 


o 




- 


400 


- °o 

^CXMoct) O o o ° ° ° 


o 


o 


o° 


o 








■ 


300 


- 
















- 


200 
100 


/■= 9.267 GHz 
MCCL Single Crystal 
_ W II Long Growth Axis 




1 








1 




°o 

1 


°o_ 



50 



100 



150 200 

T(K) 



250 300 



Figure 5.8: The EPR linewidth vs. temperature at a frequency of 9.267 GHz with the field 
parallel to the long growth axis of a MCCL single crystal (w = 3.2 mg). 



137 

245 K is evident as a sharp change in the linewidth. In addition, below 50 K, the linewidth 
increases dramatically, indicating that the magnetic exchange is also increasing. The low 
temperature linewidth increase combined with the nearly constant Lande g factor suggest 
that the low temperature transition, observed in the neutron dififraction experiments, may 
be produced by magnetic interactions rather than a change in the crystal structure. 

The room temperature angular dependence of the Lande g factor in the plane 
perpendicular to the long growth axis of a MCCL single crystal (/w = 3.2 mg) is shown in 
Fig. 5.9. The solid line represents the best fit to a dipole angular dependence, 

g = 2.0176 + 0.04725 sin (0) , (5.2) 

where is the angle between the magnetic field and the crystal face. Because the crystal 
structure MCCL is monoclinic at room temperature with an angle of 97.5° between the 
a-axis and c-axis, this rotation is most likely occurring in the a-b plane. Otherwise, the g 
factor coupling would not be a symmetric fimction of the angle. 

In summary, EPR measurements confirm the presence of a 245 K structural 
transition in MCCL. The increase in linewidth combined with a nearly constant value for 
Lande g factor below 50 K suggest that the low temperature transition observed in the 
neutron diffraction experiments is produced by magnetic exchange. In addition, the room 
temperature angular dependence in the a-b plane is given by Eq. 5.2. 



138 





^.vo 


' 


' 1 ' 1 


1 1 1 .,.,., . 
MCCL Single Crystal - 




2.06 


hfi 


^^ 


H Long Growth Axis - 
H Rotation Axis 


o 


2.04 


-/ 


^ H 


H r=298K 


o 

(0 




-/ 




VoH f= 9.267 GHz 


•2= 

C3) 


2.02 


4<H 




K>\ hH 


(D 








\ 


■o 


k-\ 




\ 


(0 


2.00 


- 




\^ t^- 


_1 




- 




\h<H / ■ 




1.98 


_ O 


Experimental 


Data \, , yf^ - 








- Fit to Eq. 5.4 with >=2d^^ 




- 




1.96 


i 


g = 2.0176 + 
.1.1 


0.04725 sin(0) H>-^ 
1 1 1 1 1 1 1 1 1 1 



20 40 60 80 100 120 140 160 180 

Angle (degrees) 



Figure 5.9: The room temperature Lande g factor vs. angle at a frequency of 9.267 GHz 
in the plane perpendicular to the long growth axis of a MCCL single crystal (/w = 3.2 mg). 
The solid line represent a best fit to a sinusoidal curve, Eq. 5.2. The horizontal error bars 
represent an estimate of the uncertainty in the angular adjustment. 



139 

5.3 Low Field Susceptibility Measurements 
Low field magnetic susceptibility measurements were performed using a 
commercial Quantum Design SQUID magnetometer. Figure 5.10 shows the molar 
magnetic susceptibility of MCCL powder (m = 188.5 mg) as a function of temperature in a 
field of 0. 1 T. The solid line is a fit to the data using the S=l/2 Curie equation. 



;^ = . + ^^/^\ (5.3) 



where Na is Avagadro's number and the Lande g factor. Curie temperature, 0, and 
diamagnetic constant, Xdiam, are fitting parameters. The number of moles, n, is set by the 
mass and the molar weight of 216.00 g/mol. Similar fits have been performed on single 
crystal samples parallel (m = 30.3 mg), Fig. 5.11, and perpendicular to the long growth 
axis (m = 23.5 mg). Fig. 5.12. A slight deviation fi-om Curie behavior is observed below 
10 K indicating the onset of antiferro magnetic behavior observed in the EPR 
measurements. In all three cases, the Lande g factor and the Curie temperature were 
determined to be approximately g = 2.2 + 0.1 and = 3.0 + 0.1 K, respectively. The 
fitting parameters are sensitive to the size of the diamagnetic contribution, Xdiam, and 
including this parameter in the fit of the susceptibility data does not adequately establish its 
value. An alternative method of determining the diamagnetic contribution fi-om the 
susceptibility data is to plot the susceptibility multiplied by temperature, xT, vs. 
temperature and to adjust the value of Xdiam until the high temperature data 
(50K<r<250 K) approaches a horizontal line (see Figs. 5.13 through 5.15). For 



140 



0.10 



o 




"I ' 1 ' 1 ' 1 ' — 

Experimental Data 
S= 1/2 Curie Law 

n= 1 

g = 2.18 + 0.01 

= 3.00 + 0.05 K 

= (-7.2 + 0.1)x10"'emu/mol 



MCCL Powder 
H=0.1 T 

■ 0|0 OOP O i O o o O i Q Q - 



200 



250 



300 



Figure 5.10: The molar magnetic susceptibility of MCCL powder (m = 188.5 mg) as a 
function of temperature in a field of 0.1 T. The solid line is a Curie fit to the data with the 
Lande g factor. Curie temperature, and Xdmm as fitting parameters (see Eq. 5.3). 



141 



0.10 



0.08 




"1 ' 1 ' 1 ' 1 '- 

Experimental Data 
S= 1/2 Curie Law 
n= 1 

fif = 2.22 + 0.01 
= 2.87 + 0.05 K 

= (-5.5+ 1.2)x10"'emu/mol 



MCCL Single Crystal 

H = 0.1 T II Long Growth Axis 

|0 O 9 O , Q OOP o , o o 



200 



250 



300 



Figure 5.11: The molar magnetic susceptibility of a MCCL single crystal (m = 30.3 mg) 
as a fianction of temperature in a field of 0.1 T parallel to the long growth axis. The solid 
line is a Curie fit to the data with the Lande g factor. Curie temperature, and Xdiam as fitting 
parameters (see Eq. 5.3). 



142 



0.10 



0.00 



1 — ' — \ — ' — r 

o Experimental Data 
S= 1/2 Curie Law 
n = ^ 

g = 2.13 + 0.01 
= 3.22 + 0.06 K 




= (-9.8 + 1 .2) X 1 0"* emu/mol 



H = 0.1 T 

H 1 long growth axis 



50 



^-e I o . o ^ ^ ^ , - -] A ^ ^.^ ^ 
100 150 200 250 300 



T(K) 



Figure 5.12: The molar magnetic susceptibility of a MCCL single crystal (m = 23.5 mg) 
as a function of temperature in a field of 0.1 T perpendicular to the long growth axis. The 
solid line is a Curie fit to the data with the Lande g factor. Curie temperature, and Xdiam as 
fitting parameters (see Eq. 5.3). 



143 



0.40 



0.15 



T 1 1 r 




I 

Structural 
Transition 



1 ' — 



IVICCL Powder 
H=0.1 T 



± 



± 



50 100 150 

T(K) 



200 



250 



300 



Figure 5.13: The molar magnetic susceptibility multiplied by temperature for MCCL 
powder (m = 188.5 mg) as a function of temperature in a field of 0.1 T. The diamagnetic 
contribution was determined by adjusting an additive constant until the high temperature 
data approached a horizontal line. The Landee g factor was then calculated fi-om the high 
temperature 'x,T value. A sudden change in the paramagnetic behavior at approximately 
240 K is attributed to a structural transition. 



144 



0.44 



0-40 h ^^CJEIEDOCP^ 

0.36 '-i 




^uoooooooooooooeo 



oO°°° ^ 



\ 



Structural 
Transition 



IVICCL Single Crystal 

H = 0.1 T II Long Growth Axis j 



J I L 



50 100 150 200 

T(K) 



250 



300 



Figure 5.14: The molar magnetic susceptibility multiplied by temperature for a MCCL 
single crystal {m = 30.3 mg) as a function of temperature in a field of 0.1 T parallel to the 
long growth axis. The diamagnetic contribution was determined by adjusting an additive 
constant until the high temperature data approached a horizontal line. The Lande g factor 
was then calculated from the high temperature xT value. A sudden change in the 
paramagnetic behavior at approximately 220 K is attributed to a structural transition. The 
feature at a temperature of approximately 50 K may be produced by magnetic ordering. 



145 




1 ' r 



T 







q(5UL)w<-'i„'0 OOOOOuUU00U(5Uo- 



Structural 
Transition 



MCCL Single Crystal 

AV = 0.1 T 1 Long Growth Axis J 

I I I I.I. 



50 



100 



150 

T(K) 



200 



250 



300 



Figure 5.15: The molar magnetic susceptibility multiplied by temperature for a MCCL 
single crystal (m = 23.5 mg) as a function of temperature in a field of 0.1 T perpendicular 
to the long growth axis. The diamagnetic contribution was determined by adjusting an 
additive constant until the high temperature data approached a horizontal line. The Lande 
g factor was then calculated fi-om the high temperature xT value. A sudden change in the 
paramagnetic behavior at approximately 200 K is attributed to a structural transition. The 
feature at a temperature of approximately 50 K may be produced by magnetic ordering. 



146 

MCCL, the g value is changing above 250 K and consequently the xT value is only 
approximately constant between 50 K and 250 K. In addition, the Lande g factor can be 
estimated by the value of this horizontal line. Moreover, the Curie temperature can be 
determined by refitting the susceptibility data with g and Xdiam as fixed parameters. The 
values of all three parameters for powder and single crystal MCCL are listed in Table 5.2. 
These values can be compared with the results of the EPR measurements. Table 5.1. 
From Fig. 5.13, two interesting features are apparent. First, the xT value approaches the 
horizontal line fi-om below, indicating antiferromagnetic behavior. Second, the data 
deviates fi-om the horizontal line at approximately T ~ 240 K due to the previously 
discussed structural transition. In addition. Figs. 5.14 and 5.15 show a small bump in the 
data at r ~ 50 K possibly due to the transition observed at the same temperature in the 
neutron diffraction experiments, which is most likely magnetic in nature. This feature is 
also evident in the powder susceptibility, but is difficult to observe. 



Table 5.2: The Lande g factor, diamagnetic contribution, Xdiam, and the antiferromagnetic 
Curie temperature, 0, determined by fitting the susceptibility data. 





Powder 


//|| Long 
Growth Axis 


//I Long 
Growth Axis 


mass(g) 


188.5 


30.3 


23.5 


g fi-om xT 


2.01 ±0.05 


2.1 ±0.1 


2.0 ±0.1 


Xdiam (emu/mol) fi-om yj 


(-7±l)xl0"^ 


(-6±l)xl0-' 


(-4±l)xlO^ 


0(K) fi-om X 


2.11 ±0.03 


2.28 ± 0.02 


2.51 ±0.02 


g fi-om slope of 1/x 


2.013 ±0.002 


2.1 18 ±0.003 


1.998 ±0.001 



147 

Plotting the inverse susceptibility vs. temperature is another useful method for 
viewing the susceptibility data. Figures 5.16 through 5.18 show the inverse molar 
magnetic susceptibility for MCCL powder and single crystal samples as a function of 
temperature in a field of 0.1 T. The constant diamagnetic contributions, Xd'am listed in 
Table 5.2 were subtracted from the data before taking the inverse. In all three cases, the 
linear dependence on temperature is indicative of paramagnetic behavior. A deviation at 
the approximate temperature of 10 K (see inset) is ascribed to magnetic interactions. The 
value of the Lande g factor can also be determined from a linear fit to the inverse 
susceptibility data and those values are included in Table 5.2. 

Figure 5.19 shows the molar magnetic susceptibility of MCCL powder (w = 
188.5 mg) as a function of temperature in a field of O.I T. The solid line is a alternating 
chain fit to the data using the method of Chiara et al [7]. The large uncertainties in the 
parameters J\ and J2 arise due to the absence of a peak in the susceptibility. Fitting the 
data using single crystals produced similar results. Nevertheless, in spite of the large 
uncertainty, the data provide evidence that the exchange constants are on the order of 
~IK. 

Low field (// < 5 T) magnetization experiments were also performed on powder 
and single crystal samples of MCCL (Figs. 5.20 through 5.22) at a temperature of 2 K. 
The solid lines are a fit to the iS = 1/2 Brillouin fiinction 



M=l«iV,g//,tanh[^^^j. (5.4) 



148 



800 



600 



3 

E 

(D 

^ 400 



E 




200 - 



300 



Figure 5.16: The inverse molar magnetic susceptibility for MCCL powder (m = 
188.5 mg) as a function of temperature in a field of 0.1 T. A constant diamagnetic 
contribution of Xdiam = -7 x 10~^ emu/mol was subtracted from the data before taking the 
inverse. The deviation fi-om paramagnetic behavior (arrow) at a temperature of 
approximately 8 K is attributed to magnetic exchange. 



149 



Z5 

£ 

o 

£ 




300 



Figure 5.17: The inverse molar magnetic susceptibility for a MCCL single crystal (m = 
30.3 mg) as a fimction of temperature in a field of 0.1 T. A constant diamagnetic 
contribution of Xd,am = -6 x 10"^ emu/mol was subtracted fi-om the data before taking the 
inverse. The deviation fi-om paramagnetic behavior (arrow) at a temperature of 
approximately 9 K is attributed to magnetic exchange. 



150 



800 

700 

600 

E 500 


O 400 

^ 300 

200 

100 





T 



-1 1 1 1 1- 

o Experimental Data 
S= 1/2 Curie 

slope = 2.672 + 0.003 (mol/K emu) 

0= 1.9 + 0.2 K 



MCCL Single Crystal 

H=0.1 T 

H 1 Long Growth Axis 




300 



Figure 5.18: The inverse molar magnetic susceptibility for a MCCL single crystal (m = 
23.5 mg) as a function of temperature in a field of 0.1 T. A constant diamagnetic 
contribution of Xd.am = -4x10"^ emu/mol was subtracted from the data before taking the 
inverse. The deviation from paramagnetic behavior (arrow) at a temperature of 
approximately 10 K is attributed to magnetic exchange. 



151 



0.10 



0.00 



o 




-1 1 1 1 1 1 r 

Experimental Points 

Alternating Chain Fit 

J^ = 2.8+ 1.4 K 

J^ =2.3+ 1.6 K 

Inpurity Cone. = 0.2 + 0.2 % 

g = 2.01 



MCCL Powder 
H = 0.1T 

^- €> 0,0 OOP o , o OOP Q . Q o 



50 



100 



150 

T(K) 



200 



250 



300 



Figure 5.19: The molar magnetic susceptibility of MCCL powder {m = 188.5 mg) as a 
function of temperature in a field of 0.1 T. The solid line is an alternating chain fit to the 
data using the method of Chiara et al. [7]. The large uncertainties in the parameters J\ 
and Ji arise due to the absence of a peak in the susceptibility. Fitting the data using single 
crystals produced similar results. 



152 



3000 



2500 



^ 2000 



E 
CD 

E 



1500 



1000 



500 - 



— 1 1 1 1 — 

O Experimental Data 

S = 1/2Brillouin 

n = 0.713 + 0.004 
g = 1.639 + 0.006 




MCCL Powder 
7=2K 



H(T) 



Figure 5.20: The molar magnetization for MCCL powder (m = 188.5 mg) as a function of 
field at a temperature of 2 K. The solid line is an 5 = 1/2 Brillouin fit to the data with a 
value for the Lande g factor of 1.639, which disagrees with the value fi-om the EPR and 
susceptibility data, demonstrating that at this temperature, the magnetic behavior is not 
paramagnetic. 



153 



3500 
3000 
2500 
2000 



o 
E 

O 



E 1500 




—I I I I r 

O Experimental Data 
S= 1/2Brillioun 

n=^ 

g= 1.401 +0.004 



1 r- 




MCCL Single Crystal 

7=2K 

H II Long Growth Axis 

J 1 I I 1^ 



2 3 4 5 

H(T) 

Figure 5.21: The molar magnetization for a MCCL single crystal (m = 30.3 mg) as a 
function of field parallel to the long growth axis at a temperature of 2 K. The solid line is 
anS= 1/2 Brillouin fit to the data with a value for the Lande g factor of 1.401, which 
disagrees with the value fi-om the EPR and susceptibility data, demonstrating that at this 
temperature, the magnetic behavior is not paramagnetic. 



154 



3000 


- O 


1 ' 1 ' 

Experimental Data 


I'll 




• 


-S = 


1/2 Brillioun 


..<^s^ 


2500 


- 


n = 


= 1 


M^^'^ ■ 






9 = 


= 1.329 + 0.003 


CiJ^ 


-— V 








oP^ 


"5 2000 
3 1500 


- 




G 


y^ 


• 




oJ^ 


• 


= 
0) 
^ 1000 


- 


J" 


/ 


- 


500 


y 


/ 




H 1 long growth axis - 
7=2K 



0( 







H(T) 



Figure 5.22: The molar magnetization for a MCCL single crystal (m = 23.5 mg) as a 
fimction of field perpendicular to the long growth axis at a temperature of 2 K. The solid 
line is an 5 = 1/2 Brillouin fit to the data with a value for Lande g factor of 1.329, which 
disagrees with the value fi-om the EPR and susceptibility data, demonstrating that at this 
temperature, the magnetic behavior is not paramagnetic. 



155 

In all three cases, a Brillouin function was unable to adequately describe the data using 
reasonable values for the Lande g factor. In addition, an 5 = 1 Brillouin fit or a 
combination of iS = 1/2 and S = 1 Brillouin fits provided similar results. This disagreement 
confirms that magnetic exchange is important below 5 K. Fitting the low field 
susceptibility results using the exact diagonalization method wiU not determine the 
exchange constants accurately for the reasons discussed in Chapter 4. However, by 
performing the magnetization experiments in high field and low temperature, the exchange 
constants may be determined more accurately. Consequently, the high magnetic field 
(H<30 T) studies were initiated. In summary, the susceptibility data indicates 
paramagnetic behavior down to T « 10 K, providing an estimate for the magnitude of the 
antiferromagnetic exchange constants of J~ 2 K. Furthermore, signatures of transitions at 
near 245 K and 50 K, which have been observed in the neutron scattering experiments, are 
detected in the low field susceptibility data. 

5.4 High Field Magnetization Measm-ements 
High field (0 < H < 30 T) magnetization measurements on MCCL powder were 
performed using a vibrating sample magnetometer at the NHMFL. Figure 5.23 shows the 
magnetization of MCCL powder (m = 200.8 mg) normalized to the saturation 
magnetization as a fimction of field at a temperature of 4.48 K. A small diamagnetic linear 
dependence was subtracted fi-om the data, Xd.am = -1.196 x 10"^ emu/mol. Again, the 
curve cannot be adequately fit using a Brillouin fiinction, ruling out paramagnetic behavior 
at this temperature. In fact, the saturation magnetization, Ms, is not achieved until 
approximately Hs = 25 T. The high field value needed to reach saturation 



156 

{gfiBHslka) » 33 K is particularly puzzling when compared to the proposed magnitude of 
the exchange constants J ~ 2 K from the low field susceptibility data. The solid line in 
Fig. 5.23 indicates a fit to the magnetization data using the 12 spin exact diagonalization 
technique and an alternating chain Hamiltonian, Eq. 5.1 with the best fit parameters of Ji = 
4.65 K and J2 = 4.65 K. The fit obviously does not match the data. 

The disagreement between theory and experiment worsens as the temperature is 
lowered. The magnetization of MCCL powder {m = 200.8 mg) as a function of field at a 
temperature of 1 .59 K is shown in Fig. 5.24. A small diamagnetic linear dependence was 
subtracted from the data, ^diam = -1.196 x 10"^ emu/mo 1, which was determined by fitting 
the magnetization curve at this temperature above 25 T. There are two distinct first 
derivative changes at 3.2 T and 16 T, with an approximately linear dependence on field 
between those points. These slope changes are easily identified in the first derivative of 
the data (inset). This behavior is not particular to this sample, and the first transition at 
3.2 T is apparent in Figs. 5.21 and 5.22 at a temperature of 2 K. The solid line in 
Fig. 5.24 is a fit to the alternating chain model using the exact diagonalization technique 
demonstrating the complete disagreement with the theoretical predictions. Again, both of 
the exchange constants are the same, Ji = J2 = 3.65 K. Obviously, the alternating chain 
Hamiltonian, Eq. 5.1, does not adequately describe the magnetic exchange in MCCL. 

The shape of the magnetization curve in Fig. 5.24 suggests that the magnetization 
may be explained by a combination of a Brillouin fimction and an alternating chain 
magnetization curve. Fitting the data to a combination of two curves is difficult due to the 
number office parameters. For instance, the percentage of free spin, the alternating chain 
parameters, J\ and Jj, and the Lande g factors are unknown. An attempt was made to fit 



157 



the low temperature magnetization assuming a Lande g factor of 2.0 for all spins. The 
values of Ji and J2, as well as the percentage of free spins were varied to obtain the fit 
shown in Fig. 5.25. By subtracting a Curie contribution, representing 47% of the total 
spins, from the powder susceptibility curve (Fig. 5.10), we obtain the result shown in 
Fig. 5.26. The curve in Fig. 5.25 is similar to the ladder susceptibility curves presented for 
BPCB (see Fig. 4.6). The results, while not conclusive, suggest that MCCL is most likely 
a combination of free spins and a low dimensional magnetic system. 

In conclusion, a monoclinic to triclinic structural transition in MCCL has been 
observed in neutron diffraction experiments at 245 ± 5 K. A second transition at 
approximately 50 K, in the neutron diffraction experiments, may be attributed to magnetic 
ordering on the basis of EPR measurements. The susceptibility as a fimction of 
temperature indicates that the antiferromagnetic exchange becomes important below 10 K. 
The low temperature (T < 10 K) magnetization measurements cannot be adequately fit 
using a Brillouin fimction, Eq. 5.4, confirming the importance of magnetic exchange. 
However, the magnetization data cannot be explained using an alternating chain model 
alone. In addition, at the lowest temperature, T = 1.59 K, the magnetization has two 
distinct changes in slope at 3.2 and 16 T, with a linear dependence on field between these 
field values. The data are most likely explained by a combination of free spins and spins 
involved in magnetic exchange with a low dimensional geometry. An initial "guess" of the 
exchange pathways in MCCL at low temperature requires the crystal structure. Further 
neutron diffraction experiments at low temperature (r< 10 K) will help to clarify details 
of the magnetic exchange. 



158 



W 



1.0 



0.8 



0.6 



0.4 - 



0.2 - 



.miiiii I I I 



0.0 








Experimental Data 
Exact Diagonal ization Fit: 
J^ = 4.65 + 0.05 K 
X = 4.65 + 0.05 K 



10 



15 
H(T) 



20 



25 



30 



Figure 5.23: The magnetization of MCCL powder (m = 200.8 mg) normalized to the 
saturation magnetization as a function of field at a temperature of 4.48 K. A small 
diamagnetic linear dependence was subtracted fi-om the data, Xdiam =" 
-1.196x10^ emu/mol. Saturation magnetization is not achieved until approximately 
25 T. The solid line is a fit to the magnetization data using an alternating chain model and 
the exact diagonalization technique. 



159 



w 



1.0 


—I — 1 — 1 — I — p-i — 1 — 1 — r— 

_ MCCL Powder 
7= 1.59 K/^ 


1 1 1 1 


1 ' 


1 1 1 1 1 1 1 1 1 1 1 1 


o 


Experimental Data 






y 




- Exact Diagonalization Fit: 






0.8 


- / 


/ 


J^ = 3.65 + 0.05 K 




1 /y 




J^ = 3.65 + 0.05 K 


0.6 


^ ,^^ 




\ 




— 






'P 0.15 ' 




o 










I 


- o 
o 






0.4 


~ / 


>.0.10 

to 


o 

_ o 

o 




- 




/ 


2 


■ \ 








/ 
/ 


1, 0.05 


- v 




" 




o / 


T3 








0.2 


_ o / 
/ 




1 1 11 1 1 




— 




-° / 


5 


10 15 20 25 30 - 


n n 


/ 1 1 1 1 1 1 1 1 1 


1 1 1 1 1 1 1 


H(T) 
1 1 1 1 1 1 1 1 1 1 1 1 







10 



15 

H(T) 



20 



25 



30 



Figure 5.24: The magnetization of MCCL powder (m = 200.8 mg) as a function of field 
at a temperature of 1.59 K. A small diamagnetic linear dependence was subtracted fi^om 
the data, Xdiam = -1.196 x 10" emu/mo 1, which was determined by fitting the above 
magnetization curve above 25 T. The magnetization has two distinct first derivative 
changes at approximately 3 T and 16 T with approximately linear behavior between those 
points (see inset). This behavior is not particular to this sample and the first transition at 
3.2 T is apparent in Figs. 5.21 through 5.22 at a temperature of 2 K. The solid line is a fit 
to the magnetization data using an alternating chain model and the exact diagonalization 
technique. 



160 



1.0 



0.8 



' ' ' ' I ' ' ' ' I 
I- MCCL Powder 
7=1. 59K 



CO 



0.6 



0.4 



0.2 



0.0 




1 — I — I — I — I — I — I — I — I — I — I — I — r 



Experimental Data 
47% Free Spin and 
53% Alternating Chain with 
J. = 8.55 K 



J^ = 7.40 K 



J_L 



I ■ ■ 



10 



15 

H(T) 



20 



25 



30 



Figure 5.25: The magnetization of MCCL powder (m = 200.8 mg) as a function of field 
at a temperature of 1.59 K. The solid line is a fit to the magnetization data using a 
combination of tree spin and an alternating model with Ji = 8.55 K and J2 = 7.40 K. 



161 



20 


1 1 1 1 1 I 1 1 1 1 1 1 1 1 1 1 1 1 1 




S ° ° Curie Term subtracted (47% of spins) ' 
] X o from MCCL Susceptibility 

- S ° 




? 15 


- o o 


E 


■ o °o 


— -. 


o 


3 


- o o 


E 


o 
o o 





°o 


<? 10 


- o °Oo 


o 


Oq 


^^ 


Oo 




■ ° ""- . ■ 


^ 5 


:° — -■ 


n 


, , ■ 1 , . . 1 . . . " 



10 



20 30 

T(K) 



40 



50 



Figure 5.26: The susceptibility of MCCL powder (m = 188.5 mg) as a function of 
temperature in a field of 0. 1 T. A Curie susceptibility curve calculated using 47% of the 
total spin has been subtracted fi-om the data. The shape of this curve is reminiscent of 
other low dimensional susceptibility curves (see Fig. 4.6). 



CHAPTER 6 

ZERO SOUND ATTENUATION NEAR THE QUANTUM LIMIT IN NORMAL 

LIQUID ^HE CLOSE TO THE SUPERFLUID TRANSITION 



The Fermi energy of a system of non-interacting Fermi particles in the T -> limit 
is determined only by the number of particles, i.e. each quantum state is filled until the 
Fermi energy is reached. In ^He, at finite temperature, interactions between He atoms 
mean that the energy of any particular particle also depends on the occupation of the other 
levels. Hence, the ^He atoms are termed "quasiparticles" with an effective mass m* which 
is proportional to the quasiparticle density at the Fermi energy. The propagation of sound 
through liquid ^He is sensitive to the interaction between quasiparticles and the mean time 
between collisions, r. The two primary modes of sound propagation are hydrodynamic or 
first sound (cor « 1) and collisionless or zero sound (cor » 1), depending on the 
fi-equency of the sound, o) = 2nf. At temperatures well below the Fermi energy (T« Tp), 
and above the superfluid transition temperature, 7c, the attenuation of both first and zero 
sound is well described by Landau Fermi Liquid Theory [27,28]. 

Zero sound is transmitted as a collective mode by an oscillatory deformation of the 
Fermi sphere. In the zero sound regime the attenuation is dominated by scattering within 
a continuous band of quasiparticle energies near the Fermi energy, AE = Ef± kaT, and has 
a "F dependence due to quasiparticle relaxation. At high frequencies 

{ksT « h(o« ksTF) collisions will scatter quasiparticles to unoccupied energy levels 



162 



163 

greater than ksT away from the Fermi energy. In this instance, the quantum nature of the 
Fermi states becomes significant, and this quantum scattering produces a second, quantum 
term in the attenuation of zero sound. Landau's prediction for zero sound attenuation 
[27,28], including the quantum term, is 



a,{(o,T,P)^a\P)T' 



( 



1 + 



h(o 



InkgT J 



(6.1) 



where a\F) represents the pressure dependent quasiparticle scattering near the Fermi 
surface. Although the factor a\P) is difiBcult to calculate, it has been measured at several 
pressures [92-96]. 

Because the quantum term in Eq. 6.1 is effectively temperature independent, its 
determination requires measurement of the absolute attenuation. Attempts have been 
made to verify this term [29-31], with the most recent by Granroth et al. [32]. A 
summary of previous experiments is given in Table 6.1, including the minimum 
temperature, operating pressure, frequency, and the relative value of the quantum term in 
Eq. 6.1 at the minimum temperature. In the experiment by Granroth et al. [32], the 
temperature and pressure were held fixed while the frequency was swept from 8 to 
50 MHz. To provide absolute attenuation, the received signals were calibrated against the 
attenuation in the first sound regime. The result of this measurement was that the 
frequency dependence of the quantum term was a factor of 5.6 + 1.2 greater than the 
prediction. However, the frequency range was limited by the polyvinylindene flouride 
(PVDF) transducers that were used so that fmax ~ 50 MHz. By extending the experiment 



164 

to higher frequencies, it should be possible to more accurately determine the quantum 
term. In this experiment LiNbOs transducers were used to extend the frequency range to 
approximately y^a:t ~ HO MHz. Again, absolute calibration of the attenuation was 
determined using measurements in the first sound regime. For both the zero and first 
sound data, the temperature and pressure were held fixed while received signals were 
averaged at several discrete frequencies. 



Table 6. 1 : The minimum temperature, operating pressure, frequency, and relative value of 
the quantum term in the zero sound attenuation (Eq. 6.1) at Tnun- The pressure marked 
"SVP" represents saturated vapor pressure. 



Reference 


[co/27^ 
(MHz) 


-'rrrin 

(mK) 


P 
(bars) 






1 


Matsumoto et aL [29] 


389 


2.5 


3 


0.11 


Matsumoto et al. [31] 


389 


6 


0.4 


0.25 


389 


6 


3 


0.25 


389 


7 


5 


0.18 


Barreetal. [30] 


84 


7 


SVP 


0.01 


254 


7 


SVP 


0.08 


422 


7 


SVP 


0.21 


592 


7 


SVP 


0.42 


Granroth et al. [32] 


46 


1.08 


1 


0.11 


Current Work 


23 


1.11 


1 


0.025 


66.6 


1.11 


1 


0.21 


107 


1.11 


1 


0.54 


28 


1.55 


5 


0.019 


68 


1.55 


5 


0.11 


108 


1.55 


5 


0.28 



165 

This chapter is divided into four sections, with the experimental details specific to 
the quantum limit experiment given in Section 6.1. Sections 6.2 and 6.3 provide the 
results in the zero sound and first sound regimes, respectively. The final section discusses 
the overall results of the experiment as well as possible sources of error. 

6.1 Experimental Details 

This experiment was performed using the same cylindrical sample cell employed in 
the previous experiment by Granroth et al. [32]. However, the transducers and MACOR 
spacers are different fi-om those used in the previous experiment. Details of the He 
acoustic technique including the transducers, spectrometer, cell dimensions, and 
thermometry, are described in Section 2.9 of this dissertation and have been reported 
elsewhere [39]. A typical pulse sequence consisted of 64 pulses with 4 s delay between 
each pulse. A phase cycling procedure was used for averaging over multiples of four 
pulses and a relatively long 4 ^s pulse was used to limit the fi-equency bandwidth of the 
transmitted pulses. The in-phase and quadrature components were separately digitized at 
10 M sample/s and 2048 samples were recorded for each pulse. 

The output level of the spectrometer was fixed at approximately 13 dBm 
(maximum) during the experiment so that comparisons between received signal levels 
could be performed. To avoid heating the ^He liquid, a variable inline attenuator (set at 
-20 dB) was used on the transmitter side so that the cumulative attenuation fi-om the 
cables (see Fig. 2.13) provided a pulse input power of approximately 1 nJ. To test for 
linearity, signals were averaged at fixed fi"equency and temperature using different 
attenuator values. Figure 6.1 shows the integrated amplitude vs. the inline attenuator 



166 

setting at a frequency of 22 MHz and a temperature of approximately 100 mK. The 
integrated amplitudes have been adjusted according to their attenuation so that a linear 
relationship between amplitude and input power would generate a horizontal line. From 
this analysis, it is clear that the attenuator must be set to at least -20 dB to remain in the 
linear region. Using further attenuation would lower the signal size unnecessarily and 
consequently the inline attenuator setting was -20 dB throughout the experiment. 

The experiment is divided into two separate stages with the sample cell pressure 
fixed at either 1 or 5 bars. The cell pressure was changed using a standard zeolite 
absorption ^He pressure bomb and was monitored by a Digiquartz transducer, whose 
calibration is discussed in Section 2.9. After the desired pressure was set, a room 
temperature valve was closed at the top of the cryostat. For this experiment, the pressure 
was not measured in situ. The volume of the capillary line between the top of the cryostat 
and the ^He cell will produce a small change in the cell pressure which must be considered 
in the error analysis (see Section 6.4). 

For each pressure, approximately 12 fixed temperatures were used in the zero 
sound regime and 20 fixed temperatures in the first sound regime. The order in which the 
data was taken, as well as a list of the temperatures, pressures, and frequencies for each 
data set are supplied in Appendix I. During the experiment, at the pressure of 1 bar, it 
was determined that the received signals at 19, 20, and 21 MHz entered a non-linear 
regime, as those frequencies are close to the transducer resonance. It is not clear if the 
non-linear behavior is a resuh of overdriving the transducer or imparting too much energy 
into the liquid. A graph of the integrated amplitude as a fianction of the inverse square of 
temperature at 21 MHz and 1 bar is given in Fig. 6.2. A distinct change in slope is 



167 



3 

■a 

3 
*^ 

"5. 

E 
< 

"D 

B 

(0 

k. 

D) 
0) 



■o 

0) 

0) 

13 

< 



23.6 



23.4 - 



23.2 - 



23.0 - 



22.8 - 



22.6 



1 ' 


' 1 ' 


' 1 

A 


1 

A 


- 


A 




- 


- 






f=22MHz ■ 


— 






P = 5 bars - 


A 
1 


1 


1 1 


7«100mK . 

■ ill 



10 13 16 19 

Inline Attenuator Setting (dB) 



Figure 6.1: The integrated amplitude vs. the inline attenuator setting at a frequency of 
22 MHz and a temperature of approximately 100 mK. The integrated amplitudes have 
been adjusted according to the inline attenuator value so that a linear relationship between 
amplitude and input power would generate a horizontal line. Before integration, the 
electrical feedthrough was subtracted. 



168 

apparent corresponding to the beginning of non-linear behavior. Consequently, the data at 
the frequencies of 19, 20, and 21 MHz are not included in the analysis. The integrated 
amplitude vs. frequency indicating the transducer response at 1 bar and a temperature of 
1.109 mK is shown in Fig. 6.3. The large relative transducer response near 21 MHz is 
evident. Although frequencies near the transducer resonance at 21 MHz can be used, the 
inline attenuator must be changed during the experiment. Since the inline attenuator is not 
designed to provide precise attenuation, it is difBcult to make absolute comparisons of the 
signal levels, especially at the level required for this work. The frequency of 20 MHz 
provided reasonable results at 5 bars and therefore was included. Before and after the 
acquisition of each data set, the melting curve thermometer value was recorded and 
averaged to account for any temperature drift. Between frequencies, there was at least 8 
minutes delay to avoid heating the liquid. 

The received amplitude of a 16 MHz pulse as a fimction of time at a pressure of 
1 bar and a temperature of 1.11 mK is shown in Fig. 6.4 (A). The coherent noise due to 
electrical feedthrough is evident in the first 10 fxs of data. To improve the signal to noise 
ratio, a reference signal was taken at 3.52 mK (Fig. 6.4 (B)), where the received pulse 
amplitude cannot be differentiated from the noise, and was subsequently subtracted from 
all of the zero sound data at 1 bar. The real and imaginary portions of the signal were 
subtracted separately before calculating the magnitude. The result after subtraction is 
shown in Fig. 6.4 (C) where the crosstalk feature at approximately 7 ^s has clearly been 
removed. Prior to any fiirther analysis, a similar procedure was performed for the zero 
sound data at 5 bars and the first sound data at both pressures. 



169 



:3 



1 ( 


-v ' 1 ' 1 


' 1 


16 


— ^i» 


— 


15 


X,,^^ 


- 


14 


- 


^"\ - 


13 


P = 1 bar 
f=2^ MHz 

1 1 r 1 


1 



100 



200 

1/T'(1/k') 



300 



Figure 6.2: The logarithm of the integrated amplitude vs. the inverse square of 
temperature at 1 bar and 21 MHz. The solid line is a guide to the eye. A distinct change 
in slope is apparent due to non-linear behavior at this frequency. 



170 




resolution threshold 



20 40 60 80 

Frequency (MHz) 



100 



Figure 6.3: The integrated amplitude vs. frequency indicating the transducer response at 
1 bar and a temperature of 1.109 mK. The transducer resonances a 19 and 21 MHz are 
evident. The solid lines are a guide to the eye. The dashed lines indicate the non-linear 
and resolution thresholds. 



171 



3 

■D 

3 



50 

25 


50 

25 


50 

25 





Ph 



ivfY>- 



f=16MHz A 

P=1bar 

7=1.11mK 

\A/ithout crosstalk subtracted 



W^ 



f=16MHz 
P=1bar 
T= 3.52 mK 
crosstalk only 



B - 



K 



/ 



i^.A» 



^loa^^W 



f=16MHz 

P=1bar 

r=1.11mK 

with crosstalk subtracted 



C - 



^tllfff^^^^ 



10 15 20 25 30 35 40 45 50 

Time (^s) 



Figure 6.4: The received amplitude of a 16 MHz pulse as a function of time at a pressure 
of 1 bar and a temperature of 1.11 mK (A). The coherent noise due to electrical 
feedthrough is evident in the t5rst 10 i^s of data. At a temperature of 3.52 mK (B) the 
signal is attenuated and only the crosstalk remains. When the data from (A) is subtracted 
from (B) the result is (C). 



172 

The absolute attenuation is linearly dependent on the path length, and therefore an 
in situ measurement of this length is desirable. The response observed from a 22 MHz 
pulse at 5 bars and 1.55 mK is shown in Fig. 6.5. The time length between successive 
echoes, combined with the tabulated velocity of zero sound from Halperin and Varoquaux 
[61], allow us to calculate a path length, /, of 

/ = 0.322 ± 0.002 cm. (6.2) 

The uncertainty in this measurement is determined by the 100 ns time resolution of the 
spectrometer. Knowing the path length and the time delay before each received pulse, we 
can calculate the velocity as a fiinction of temperature as shown in Fig. 6.6 for the 
pressure of 1 and 5 bars. The time delay was determined by the sudden increase of the 
sound amplitude corresponding to the arrival of the pulse in the 23 MHz data. The 
23 MHz data was chosen because the start of the pulse is easy to identify. Received 
pulses with frequencies above approximately 60 MHz appear rounded because the higher 
frequency harmonics of the pulse are attenuated. Within the uncertainty, the velocity is 
constant at both pressures. The error bars have been determined by a combination of the 
uncertainty in the path length and the uncertainty in the time of arrival, which is essentially 
one-half the time resolution of the spectrometer. In this experiment, a room temperature 
valve was used to access the sample space (see Section 2.9) and consequently, the 
pressure was slightly temperature dependent due to the small volume of the ^He capillary. 
At 1 and 5 bars, the pressure change from 100 to 800 mK is approximately 0.1 bars and 
0.2 bars, respectively. The pressure change was not measured directly during the 



173 



240 
210 



^ lU r 
180 - 



-> 1 ' r 

25.80 + 0.05 US 



d 150 - 

^ 120 h 

^ 90 IK 

Q. 

I 60 IK 



30 - 

^ 



W^ 



P = 5 bars 
T= 1.55 mK 
f=22MHz 




20 40 60 

Time (n,s) 



80 



100 



Figure 6.5: A 22 MHz received pulse vs. time at 5 bars and 1.55 mK. The time length 
between successive echoes, combined with the velocity of zero sound measured elsewhere 
[61], allow us to calculate a path length of 0.322 + 0.002 cm. 



174 



200 



196 



192 



250 



^ 245 

CO 






240 



235 



"1 <- 



1 <- 



I 



-■ — r 



Volume = 35.6516 cm 
P » 1 bar 



9 ^im9fi.g 



T ^ T 



'±'"i- 



t^ 



'"'"..».»». 



o Experimental Data 
' Roach et al. [97] 

Abratiam etal. [99] 

■ Halperjn and Varoquaux [61] 



I I I I I I I I 



+ 



I I I 



-+- 



-h-H 



■'"1A», 



r 



■"""*.,», 



Volume = 32.5883 cm 
P « 5 bars 



QO Q 6 



t*"44.^, 



o Experimental Data 
' Roacti et al. [97] 

Abratiam etal. [99] 

■ Halperin and Varoquaux [61] 



_L 



_L 



_L 



J_ 



100 200 300 400 500 

T(mK) 



600 



700 



800 



Figure 6.6: The velocity of first sound as a fiinction of temperature at a pressure of 1 and 
5 bars calculated using the path length and the measured signal delay for a 23 MHz pulse. 
The triangles are calculated fi"om the expression given by Roach et al. [98] using the 
adiabatic compressibility and density values fi-om Kollar and Vollhardt [97]. These 
theoretical curves have been adjusted by a constant (-8.00 m/s for 1 bar and -1.55 m/s for 
5 bars) to match the experimental velocity value at 50 mK. The solid lines represent the 
theoretical first sound velocity calculated using the expression given by Abraham et al. 
[99] and the density values fi"om Kollar and Vollhardt [97]. Likewise, these theoretical 
curves have been adjusted by a constant (65 m/s for 1 bar and -1.50 m/s for 5 bars) to 
match the experimental velocity value at 50 mK. All of the theoretical curves assumed a 
constant volume (see text). The first sound velocity at 250 mK given by Halperin and 
Varoquaux [61] is also shown (squares). Subsequent calculations used a constant velocity 
of 19752 cm/s for 1 bar and 24584 cm/s for 5 bars. 



175 

experiment. However, because the volume of the ^He capillary is smaD compared to the 
volume of the cell, we can assume that the cell volume will remain constant. Setting of the 
^He pressure was performed at the temperature of 250 mK. The volumes corresponding 
to 1 and 5 bars at 250 mK, determined using the results of Kollar and Vollhardt [97], are 
35.6516 and 32.5883 cm^, respectively. Using these constant volumes, the temperature 
dependence of all other thermodynamic quantities can then be calculated [97]. In Fig. 6.6, 
the triangles are calculated from the first sound velocity expression given by Roach et al. 
[98] using the adiabatic compressibility and density values from Kollar and Vollhardt [97]. 
These theoretical curves have been adjusted by a constant (-7.05 m/s for 1 bar and 
-0.35 m/s for 5 bars) to match the experimental velocity value at 50 mK. The solid lines 
represent the theoretical &st sound velocity calculated using the expression given by 
Abraham et al. [99]. Likewise, these theoretical curves have been adjusted by a constant 
(1.85 m/s for 1 bar and -0.32 m/s for 5 bars) to match the experimental velocity value at 
50 mK. The first sound velocity at 250 mK given by Halperin and Varoquaux [61] is also 
shown (squares). Subsequent fits involving the first sound velocity used the constants 
19752 cm/s for 1 bar and 24584 cm/s for 5 bars. The integration of the received pulses 
was performed over only a 4 ^is area. Furthermore, the error bars for each integration 
were determined by integrating a 4 ^s area located at / = 200 ^s, near the end of the data 
acquisition. 

The measurement of attenuation is restricted by our input power and cell size. 
Figure 6.7 shows the attenuation as a function of the logarithm of temperature at a 
pressiire of 1 bar and a frequency of 1 6 and 64 MHz. The attenuation on the left and right 
side of Fig. 6.7 is due to zero sound and first sound, respectively. The absolute 



176 



100 



10 



E 
u 



1 r 



0.1 



TTT| 1 1 1 — I I I I I I 1 1 1 I I I I l| 1 1 1 — I I I I 

o f=16MHz 

• f = 64 MHz '^'9'^ Attenuation Limit 

\ 



••— — • 



•• 



Low Attenuation Limit 

\ 



rOcfi^ 



P=1 bar 



J ■ ' I ■ ' 







10 100 

T(mK) 



1000 



Figure 6.7: The attenuation as a function of the logarithm of temperature for a 4 ^s pulse 
at a pressure of 1 bar and a frequency of 16 MHz. The upper horizontal line indicates the 
high attenuation limit where the signal is too small to be detected. The lower horizontal 
line indicates the signal level where changes in the attenuation are smaller than the noise. 



177 

attenuation (y-axis) has been determined by fitting the temperature dependence of zero 
sound and first sound data to a known expression (Sections 6.2 and 6.3). If the 
attenuation is smaller than approximately 0.4 cm"', then changes in the attenuation will be 
smaller than the scatter due to the noise (lower horizontal line). If the attenuation is larger 
than 16.2 cm' then the signal will not be detected (upper horizontal line) . 

6.2 Zero Sound 
The general strategy for measuring the quantum term in zero sound uses the 
temperature dependence of first sound to calibrate the signal levels and involves several 
steps. Essentially, we must fit the temperature dependence of the first sound amplitude to 
a known expression to determine the absolute attenuation in both the first and zero sound 
regimes. The process of converting the raw zero sound data into absolute attenuation is 
summarized in the flow chart shown in Fig. 6.8. In the first step (lower left), a value for 
both terms in the longitudinal viscosity must be chosen at 1 and 5 bars. Using this 
viscosity, we subtract a correction due to the walls of the cylindrical sample cell fi-om the 
first sound raw data, Eq. 6.8. Applying the wall correction first decreases the number of 
independent variables and improves the quality of the subsequent fits. Next, the 
temperature dependence of the first sound data must be fit to Eq. 6.9 to obtain the 
fi-equency dependent factor, F(fy). The fits were performed on the natural logarithm of 
the amplitude as a fiinction of temperature so that the factor, F{co), represents a constant 
vertical shift. Several terms are required to calculate the first sound attenuation including 
density, p, thermal conductivity, k(T), the isothermal and isobaric specific heat, Cv(7) and 
Cp(T), first sound velocity, ci, and finally the second viscosity t^T) [100]. The 



178 







%£ 




2 ■^ 




1" on 




hC^ 








oT s 




< -^ 


- 


%^ 








1 S 




t= 













.2 u 

. -s 

■1-; ai 
13 



3 
O 

CO 






(30 cs o\ 

3 ^^ 
-> O V- 



<i> 



^ 



179 

first sound velocity, C\, is determined directly from the data by measuring time-of-flight of 
first sound pulses. As far as we know, the second viscosity has never been calculated nor 
measured, but it is generally assumed to be small relative to the other contributions [100]. 
The other terms mentioned are calculated by Kollar and Vollhardt [97] using a consistent 
set of thermodynamic quantities. The pressure dependent factor a'{P) is determined by a 
linear fit to the natural logarithm of the zero sound data plotted as a fiinction of the square 
of temperature. Once this value and the frequency dependent factor, F(ci)) are known, the 
zero sound data can be compared to Eq. 6. 1 0, where the quantum term is the only 
unknown. 

A typical received pulse amplitude as a function of time and temperature for a zero 
sound pulse at 10 MHz and 1 bar is shown in Fig. 6.9. The crosstalk has been subtracted 
using the method discussed in Section 6.1. The decrease in amplitude is due to the i 
dependence of attenuation on temperature. At low frequency, 8, 10, and 16 MHz, the 
transducer response is broader and hence the received pulses resemble the square 
transmitted pulse. Higher frequency received pulses, /> 19 MHz, contain sharp peaks 
related to the transducer responses at that frequency [39]. For example. Fig. 6.10 is a 
three dimensional representation of the received pulse amplitude as a function of time and 
temperature for a zero sound pulse at 22 MHz and 1 bar. 

After subtracting the electrical crosstalk, the data are integrated inside the 4 [xs 
window corresponding to the received pulse. The natural logarithm of the integrated 
amplitude vs. the square of temperature at a frequency of 22 MHz and a pressure of 1 bar 
is shown in Fig. 6.11. Ignoring the quantum term, the amplitude. A, will obey the 
following equation, 



180 



A = y5exp(-a'(P)r^/), 



(6.3) 



where a'iP) represents the contribution from quasiparticle scattering. The value of a'{Pyi 
= 0.437 ± 0.005 is determined by a linear fit to the natural logarithm of the data. At most 
frequencies, the values of a'{P) agree within uncertainty, however there are slight 



30 

25 

'->20 

■5- 

^ 15 ■ 
T3 

. 4-* 

"q. 10 ■ 

E 

< 

5 



"" — I — ' — r 



1 — ' — r 



m 



iW 




— 1 — I — I — I — I- 

•T = 1.109 mK 
T = 1.199mK 
T = 1.455 mK 
T = 2.035 mK 



P = 1 bar 
f=10MHz 



^^'^MufM^ "* '"^^ 




J 



J 



5 10 15 20 25 30 35 40 45 50 

Time (^s) 

Figure 6.9: A typical received pulse amplitude as a fiinction of time and temperature for a 
zero sound pulse at 10 MHz and 1 bar. The crosstalk has been subtracted using the 
method discussed in Section 6. 1 . 



181 



P= 1 bar 
f=22MHz 




1.475 ^ 
1.779 -^ 



10 15 20 25 30 35 40 

Time (^s) 



2.035 



Figure 6.10: A three dimensional representation of the received pulse amplitude as a 
flmction of time and temperature for a 22 MHz zero sound pulse at a pressure of 1 bar. 
The crosstalk has been subtracted using the method discussed in Section 6.1. 



182 

variations due to subtle pressure changes. The value of the intercept and a'{P) at each 
frequency and both pressures is listed in Table 6.2. At higher frequencies, the data also 
obey a linear relationship with respect to the square of temperature, although the error 
bars are larger due to the decreased signal size. The natural logarithm of the integrated 
amplitude vs. the square of temperature at a frequency of 107 MHz and a pressure of 
1 bar is shown in Fig. 6.12 with a slope of a'{Pyi = 0.46 + 0.04 determined by a linear fit 
to the data. The value of a'{P) can be compared to values from previous experiments [92- 
96] and Fig. 6.13 shows the value of a'(P) plotted as a ftmction of pressure. The value of 
a'iv) is almost linearly dependent on the molar volume, v (Fig. 6.14), and the solid line in 
Fig. 6.14 represents the second order polynomial fit, 

a'iv) = 2.9664-0.25493v+0.0059270v- . (6.4) 

The volume dependence of a'iv) can be converted into a pressure dependence 
a'iP) using the pressure and molar volume expression given by Halperin and Varoquaux 
[61]. However, the curvature of a'iP) requires additional terms in the polynomial 
expression to match the experimental points, 

a'iP) = 1.5781-0.17351P+0.011892/^-4.0451xl0^P^+5.2079xl0-V. (6.5) 

The solid line in Fig. 6.13 is the fourth order polynomial, Eq. 6.5, which was derived from 
Eq. 6.4. Equation 6.5 is a conversion of Eq. 6.4 and is not a separate fit to the data. 



183 




T' (mK') 



Figure 6.1 1 : The natural logarithm of integrated amplitude vs. the square of temperature 
at a frequency of 22 MHz and a pressure of 1 bar. The value of a'iPyi = 0.437 ± 0.005 is 
determined by a linear fit to the data. At most frequencies, the values of a'iP) agree 
within error, however there are slight discrepancies due to pressure variation. 



12 
11 
11 



5 h 




-I — I — 1 — I — 1 — I — I- 
107 MHz 
Linear Fit with 
Intercept = 15.43 + 0.17 
Slope = -0.46 + 0.04 




P=1 bar 
A= 107 MHz 

J 1 1 1 L 



1 



T' (mK') 



Figure 6.12: The natural logarithm of integrated amplitude vs. the square of temperature 
at a frequency of 107 MHz and a pressure of 1 bar. The value of a'(Pyi = 0.46 ± 0.04 is 
determined by a linear fit to the data. 



184 



Table 6.2: The value of a'(P) as determined from a linear fit to the natural logarithm of 
the zero sound data vs. the square of temperature. The symbol "-" indicates that data is 
not available at this frequency. 



Freq.(MHz) 
(MHz) 


a'{P) (cm') 
P=lbar 


a'(P) (cm ') 
P=5bars 


8 


- 


0.92 ± 0.03 


10 


1.27 ±0.04 


0.81 ±0.04 


16 


1.32 ±0.02 


0.87 + 0.03 


19 


1.32 ±0.04 


- 


20 


1.35 ±0.02 


0.95 ± 0.05 


21 


1.32 ±0.04 


- 


22 


1.36 ±0.02 


- 


23 


1.345 ±0.006 


0.84 ± 0.03 


28 


- 


0.78 ± 0.03 


63 


1.36 ±0.02 


0.95 ± 0.04 


64 


1.357 ±0.006 


0.85 ± 0.02 


65 


1.34 ±0.03 


0.96 ± 0.03 


66 


1.30 ±0.01 


- 


66.6 


1.37 ±0.01 


0.88 ± 0.02 


68 


- 


0.90 ± 0.06 


107 


1.4±0.1 


0.93 ± 0.09 


108 


1.4 ±0.1 


0.99 ± 0.05 


avg. 


1.34 ±0.01 


0.89 ±0.02 



185 



E 

''e 

"a 




0.2 



0,0 



^ 


This Work (1999) 


D 


Ahe\ etal. (1966) 


A 


Lawson etal. (1974) 


X 


Naraefa/. (1981) 





Ketterson ef a/. (1975) 


+ 


Mast etal. (1980) 
-Result from Eq. 6.5 



10 



15 

P (bars) 



20 



25 



30 



Figure 6.13: The value of a'(P) as measured by several research groups [92-96]. The 
solid line is a fourth order polynomial, Eq. 6.4, which was derived from Eq. 6.3 using the 
results of Halperin and Varoquaux [61]. 



1.6 

1.4 

^ 1.2 

E 

-^ 1.0 
o 



0.8 
0.6 
0.4 
0.2 



D 
A 
X 

O 

+ 



26 



— I ' 1 ' r 

This Work (1999) 
Abel ef a/. (1966) 
Lawson et al. (1974) 
Nara etal. (1981) 
Ketterson ef a/. (1975) 
Mast etal. (1980) 

- Polynomial Fit 




28 



30 



32 



34 



36 



Molar Volume (cm /mol) 



Figure 6.14: The value of Qr'(v) as measured by several research groups [92-96] plotted as 
a function of the molar volume. The solid line is a second order polynomial fit (Eq. 6.3). 



186 

In order to accurately measure the absolute attenuation of zero sound, we must 
determine the large frequency dependence due to the transducer response (see Fig. 6.3). 
Consequently, it is necessary to compare the zero sound signal levels with the high 
temperature limit of &st sound. However, assuming that the transducers are pressure 
independent, it is also possible to measure zero sound attenuation using only zero sound at 
different pressures. If we compare the natural logarithm of the integrated amplitude for 
received signals at the pressures of 1 and 5 bars, we calculate the difference between the 
r -^ extrapolations of attenuation as 



[a(T -^0,P = \ bars) -a{T^0,P = 5 bars)] = 

[a'iP = 1 bars) - a'iP = 5 bars)] 



hco ] 



Itv kg ) , (6.6) 



where a is the attenuation. Although the absolute attenuation is not known, the left side 
of Eq. 6.6 is equivalent to the difference between the y-axis intercepts at each pressure 
when plotting the logarithm of the integrated amplitude vs. 7^ (see Figs. 6.11 and 6.12). 
The a'{P) values for the pressures of 1 and 5 bars are different by approximately 40% and 
so it should be possible to measure a difference between the y-axis 7-^0 extrapolation. 



The value for 



hco \ 



ylrckg 



determined using this method is plotted vs. the frequency squared 



in Fig. 6.15, where the solid line is Landau's prediction. The number of frequencies 
represented is less than the total number available because only frequencies used at both 
pressures provide this information. The error bars in Fig. 6. 1 5 are primarily determined by 



187 



^o 


o 


1 1 1 ■ 1 ■ 1 ■ , . 
Result using the difference 


20 


- 


between the 1 and 5 bars data 






Lanaau s rreaiciion, m — (n /k^ ) j ■ 


15 


- 


i- 


M 

"^ 10 


- 


-L 


^ 5 


- 


3: 




5: 


31 





_i 1 


1 1 1 . 1 1 1 . 1 r 



2000 4000 6000 8000 10000 12000 

Frequency^ (MHz^) 



Figure 6.15: The value of 



' hco ^" 



Ink 



vs. the square of frequency determined by comparing 



B J 



the r ^ limit of the attenuation at the pressures of 1 and 5 bars. The solid line is 
Landau's prediction. 



188 

the uncertainty in the a'{P) measurements. The disparity between Landau's prediction and 
the data is caused by either a pressure induced change in the transducer properties or a 
fimdamental change in zero sound attenuation between 1 and 5 bars. 

6.3 First Sound 
In order to accurately determine the zero sound attenuation, we must subtract the 
large frequency dependence due to the transducer response (see Fig. 6.3). Herein lies the 
greatest vmcertainty or propagation of systematic errors. Consequently, it is necessary to 
compare the zero sound signal levels with the high temperature, low attenuation limit of 
first sound. The first sound attenuation was measured from approximately 30 to 800 mK 
with both the temperature and pressure held fixed. For the 1 and 5 bars data, 21 and 27 
discrete temperatures were used, respectively. As in the previous section, low temperature 
(T ~ 30 mK), high attenuation data were subtracted from all other data to eliminate the 
contribution from electrical crosstalk. The amplitude vs. time for a typical received pulse 
at 16 MHz for two similar temperatures at 1 and 5 bars is shown in Fig. 6.16. The time 
delay difference is produced by the change in the first sound velocity due to pressure. In 
the same manner as the zero soimd procedure, the integration was performed over a 4 ^s 
window which was adjusted to compensate for the pressure changes. The noise level for 
each integration was produced by a fijrther integration over a 4 ^is region at / = 200 |as. 
The attenuation of hydrodynamic sound can be written as 



a = 



lpc\ 



-v+C 

V3 J 



+ K 



(\ P 



V^v ^pj 



2 



^^-^^^f^^ (6.7) 

3/7C, T 



189 



14 
12 
10 


' 1 


1 1 1 1 1 1 1 

P= 1 bar, 7=334 mK 

\ P = 5 bars, 7=391 mK ~ 


TO, 8 
Q. 

1^ 


- 


1 ; 

f=16MHz 


2 
n 




i*W*W*ft..;,A<SHiflUW« 


C 


1 10 


20 30 40 5 

Time (^s) 



Figure 6.16: The amplitude vs. time for a typical received pulse at a frequency of 16 MHz 
for two similar temperatures at the pressures of 1 and 5 bars. The time delay difference is 
produced by the change in the first sound velocity due to pressure. 



190 

where rj and i^ are the first (longitudinal) viscosity and second (bulk) viscosity, k is the 
thermal conductivity, p is the density, C\ is the sound velocity, and Cv and Cp are the 
specific heats at constant volume and pressure [27]. As mentioned in Section 6.1, the first 
sound velocity, Ci, is determined using the time delay between the arrival of the received 
pulse and the length of the cell. The thermal conductivity, k, and the specific heat at 
constant volume, Cv, have been measured by Greywall [101]. Using the measurements of 
Greywall, Kollar and Vollhardt [97] have calculated the density, p, and specific heat at 
constant pressure, Cp, using a consistent set of thermodynamic quantities. The 
contribution to the attenuation fi-om the thermal conductivity is expected to be significant 
as it has a linear dependence on temperature and a quadratic dependence on fi-equency. 
Therefore, a term corresponding to the thermal conductivity must be included when fitting 
the temperature dependence of first sound attenuation to a known expression. 

An additional attenuation correction, which accounts for scattering of the 
quasiparticles on the walls of the cylindrical cell [27,102], can be written as 



1/2 

«..// = X^ — ' (6-8) 



CO 

2Rc, 


'2jj' 
pco 



where R is the radius of the cell. The importance of this correction is illustrated in 
Table 6.3, where the value of the hydrodynamic attenuation as well as the attenuation 
fi-om the wall correction are listed for the fi^equencies of 1 and 60 MHz. For a fi-equency 
of 10 MHz and a temperature of 800 mK, the attenuation due to the wall scattering is 
14.2% of the hydrodynamic attenuation. To improve fitting accuracy, for all of the data 



191 

presented, the wall attenuation was subtracted from the data before fitting to the 
attenuation in Eq. 6.7. 

By combining Eq. 6.7 with an additional frequency dependent factor, we can write 
the an^litude of first sound, Ai, (recalling that the crosstalk and wall attenuation 
corrections have already been made) as 



2pc, 



4 ^ 




(l 


1 1 




-v+ap) 


+ K 






> 


3 ) 


[c. 


^j. 





(6.9) 



where F(o}) is the frequency dependence of the transducer and the related electronics. The 
factor F((o) is the only unknown quantity in Eq. 6.9 and hence is uniquely determined by 
fitting Eq. 6.9 to the data. It is important to note that the terms inside the exponential in 
Eq. 6.9 decrease as the temperature is increased but they do not become arbitrarily small. 
In other words, F((o) cannot be determined by simply taking the T ^><xi limit. Finally, 
once F(co) is determined from Eq. 6.9, the amplitude of zero sound can be written as 



A a = F{o))exp{-a'(P)T^l{\+x)} , 



(6.10) 



where x is the quantum term in zero sound attenuation that will be experimentally 
determined by comparing zero soimd and first sound. 



192 



The longitudinal viscosity, rj, has been measured by several research groups over 
various temperature ranges and pressures. Table 6.4 lists the primary results for t] in poise 
at saturated vapor pressure (SVP), 1 bar, and 5 bars, where the temperature is expressed 
in units of mK. The attenuation of first sound is linearly dependent on the value of the 
viscosity and the overall results derived from fitting the temperature dependence of the 
first sound attenuation to a theoretical curve are sensitive to this value. The natural 
logarithm of the integrated amplitude vs. the inverse square of temperature for a frequency 
of 22 MHz and a pressure of 1 bar is shown in Fig. 6.17. The solid line represents a linear 
fit over the temperature region from 70 mK to 800 mK. From Eq. 6.7, the value of the 
viscosity can be calculated from the slope of this graph. However, by finding the slope for 
various frequencies, we are able to increase the accuracy of the viscosity measurement. 
The slopes of linear fits from graphs such as Fig. 6.17, including additional frequencies, is 
plotted in Fig. 6.18 for 1 bar and Fig. 6.19 for 5 bars. The value at a particular frequency 
is linearly proportional to the frequency squared and the proportionality constant is the 
viscosity (see Eq. 6.7). The frequency range has been deliberately limited (/"< 28 MHz) to 
avoid possible contributions from the second viscosity which has an/ ^ dependence. 
Before fitting, the wall attenuation and the contribution arising from the thermal 
conductivity have been subtracted from the data. From the slope of the line in Figs. 6.18 
and 6.19, we calculate a viscosity of 77 = 2.11 / 7^ (P/mK^) at 1 bar and 
;/= 1.87 / 7^(P/mK^) at 5 bars. However, measurements [103,104] have shown that the 
first sound viscosity deviates from purely f behavior at high temperature (7 > 100 mK) 
and that a second term is needed. We assume the following power law form for the 
longitudinal viscosity. 



193 



Table 6.3: Tabulation of the attenuation and wall correction terms in order to establish the 
relative importance of the wall correction. 



T 
(mK) 


ai at 

10 MHz 

(cm') 


OTwaU at 

10 MHz 
(cm-') 


ai at 

60 MHz 

(cm') 


ffwaii at 

60 MHz 

(cm') 


awal|/( ai+ Owau) 

at 10 MHz 

(%) 


awau/( ecu awall) 

at 60 MHz 

(%) 


50 


3.33 


0.0909 


120 


0.222 


2.73 


0.186 


100 


0.995 


0.0497 


35.8 


0.122 


4.99 


0.340 


200 


0.371 


0.0303 


13.3 


0.0742 


8.18 


0.556 


300 


0.238 


0.0243 


8.58 


0.0595 


10.2 


0.693 


400 


0.185 


0.0214 


6.66 


0.0523 


11.5 


0.786 


500 


0.157 


0.0196 


5.64 


0.0481 


12.5 


0.853 


600 


0.139 


0.0185 


5.02 


0.0453 


13.3 


0.903 


700 


0.128 


0.0177 


4.61 


0.0433 


13.8 


0.940 


800 


0.120 


0.0171 


4.32 


0.0418 


14.2 


0.968 



Table 6.4: The longitudinal viscosity in poise at a saturated vapor pressure (SVP), 1 bar, 
and 5 bars, as measured by various research groups with temperature expressed in mK 
[103-106]. Viscosity values that were interpolated are represented by the symbol "^". 





t] (poise) 




SVP 


Ibar 


5 bars 


Carless etal. [105] 


2.68/T' 


2.64/T' + 


2.33/T^ 4= 


Bertinat et al.il" term) [104] 


1.94/T^ 


- 


- 


Bertinat et al. (2"" term) [104] 


4.33 X 10Vt°^' 


- 


- 


Nakagawa et al. [106] 


2.08/T^ 


1.99/T'4= 


1.62/T^ 


Blacker a/. (1^ term) [103] 


2.2 1/T' 


- 


— 


Black e/ a/. (2"' term) [103] 


2.63 X 10'Vt"^ 


- 





194 



-1 1 1 1 1 1 1 1 1 1 1 1 1 1 r 

o Data 

Linear Fit with 

Intercept = 15.792 + 0.005 
Slope = -0.01371 +0.00006 




f=22MHz 
P = 1 bar 



100 



_l I L 



200 

1/T'{1/K^) 



300 



400 



Figure 6.17: The natural logarithm of the integrated amplitude vs. the inverse square of 
temperature for a frequency of 22 MHz and a pressure of 1 bar. The solid line represents 
a linear fit over the temperature region from 70 to 800 mK. 



195 



^ 0.016 - 



0.012 - 



^ 



< 

::£, 0.008 

c 



0} 

§■ 0.004 

CO 



— I 1 1 , 1- 

o First Sound Slope 



Linear Fit 7 = 2.11 / f (P/mK") 




600 



Frequency^ (MHz^) 



Figure 6.18: The slope of linear fits to the natural logarithm of the integrated amplitude 
vs. the inverse square of temperature (see Fig. 6.17) vs. fi-equency squared at a pressure of 
1 bar. From the slope of this line, we are able to calculate a viscosity of 
7 = 2.1 1 / 7^ (P/mK^) using Eq. 6.7. 



0.012 

E 0.010 



t; 0.008 
^ 0.006 h 

c' 0.004 - 

o 

g. 0.002 
o 



CO 



0.000 



— 1 ^ 1 1 1 1 1 — 

o First Sound Slope 
Linear Fit ;; = 1 .87 / 1 (P/mK') 



P = 5 bars 



200 



400 



600 



800 



1000 



Frequency^ (MHz^) 



Figure 6.19: The slope of a linear fit to the natural logarithm of the integrated amplitude 
vs. the inverse square of temperature (see Fig. 6.17) vs. fi-equency squared at a pressure of 
5 bars. From the slope of this line, we are able to calculate a viscosity of 
7 = 1.87 / f(P/rnK}) using Eq. 6.7. 



196 



7(n = ^ + ^ , (6.11) 



where Fi, r2, and n are constants. Unfortunately, it is not possible to accurately determine 
the second constant, r2, directly from our data, and both reported measurements of this 
second term have been performed only at saturated vapor pressure (see Table 6.4). It is 
therefore necessary to use a previously measured viscosity at 1 and 5 bars that is 
consistent with our results. 

For the following calculations, the first constant, Fi, is taken from the results of 
Nakagawa et al. [106]. The second constant, F2, is taken from Bertinat et al. [104], 
extrapolated from saturated vapor pressure by multiplying by the same factor applied to 
make the first constants, Fi, equal to the value reported by Nakagawa et al. [106]. We 
are implicitly making the assumption that the pressure dependence of the first and second 
term are identical and that the temperature dependence of the second term remains the 
same, 1/(7*''^), at higher pressure. The final result for the viscosity expressed in poise (P) 



IS 



1.99 4.44 X 10"* 



n = ^;:r + —^;:^2 — (6-12) 



at 1 bar and 



1.62 3.62 x lQ-" 

rp2 <T-iO 



^7 = ^ + — ^^T^ii (6.13) 



197 

at 5 bars, where T has units of mK. We have more confidence in the value of r2 at 1 bar 
than 5 bars because the pressure is closer to SVP. We expect the second term, r2, in the 
longitudinal viscosity at 5 bars to be only a rough approximation. 

An expression for the second or bulk viscosity, Q has been derived by Sykes and 
Brooker [100] in terms of a collision integral; however in their analysis, no attempt was 
made to evaluate the expression numerically. Although it has generally been assumed that 
the second viscosity has a t^-T^ dependence, they predict that the second viscosity has a 
^~ 'f dependence. It should be noted that their calculation relies on the assumption that 
He IS well described by Fermi liquid theory which is not necessarily valid above the 
temperature of approximately 150 mK (T « TpllQ). Fortunately, the magnitude of the 
second viscosity at temperatures below 1 K should be negligible [28]. Ultimately, we 
must compare the data with the expected temperature dependence irom Eq. 6.9 to 
determine if the second viscosity can be omitted Irom the calculation. 

Figure 6.20 shows the natural logarithm of the integrated amplitude vs. 
temperature at the frequency of 23 MHz and a pressure of 1 bar. The solid line is the 
predicted curve generated using Eq. 6.9. The attenuation due to the walls, Eq. 6.8, has 
been previously subtracted irom the data. At the frequency of 23 MHz, the experimental 
curve agrees with the predicted attenuation from Eq. 6.9. Similarly for the pressure of 
5 bars at low frequency, the data also match the prediction as shown in Fig. 6.21 for the 
frequency of 20 MHz. However, at higher frequencies (/" > 63 MHz), a significant 
non-systematic disagreement is observed between the predicted amplitude and the data 
(Figs. 6.22 and 6.23). The disagreement is most likely produced by a discrepancy in the 
second term of the longitudinal viscosity or a large error in the first sound velocity C\{P). 



198 

Measurement of the temperature dependence of the viscosity at high pressure and 
temperature have not been performed. 

6.4 Error Analysis and Final Results 
There are two primary sources of error that we must consider: nonparallelism of 
the transducers and pressure uncertainty. The nonparallelism of the transducers will add 
an additional frequency dependent factor, N{o}), to the zero and first sound amplitudes, so 
the received amplitude is related to the transmitted amplitude [107,108] as 

Ar.. ^ A,^^Ni(D)e-"' . (6.14) 

The factor Nio)) is expressed in terms of a Bessel fiinction 



N((o) = 2- 



(2n-lfRQ 



(6.15) 



{2n-l)'"R@ 
c 



where is the angular error and n is the received pulse number. In a previous experiment 
[32], the nonparallelism was determined by filling the cell with "^He and measuring the 
attenuation of closely spaced frequencies from 8 to 64 MHz. A plot of the amplitude of a 
4 ^s received pulse in "He as a fimction of time and frequency at 1 bar and 30 mK is 
shown in Fig. 6.24 [109]. A sudden increase in attenuation corresponds to a zero of the 
Bessel fiinction. However, for this experiment, the transducer properties could not 



199 



-■ — r 




1 — ' — I — ^ 



_L 



Data 
Fit 

_l L 



300 400 500 600 

T(mK) 



700 800 



Figure 6.20: The natural logarithm of the integrated amplitude vs. temperature at the 
frequency of 23 MHz and a pressure of 1 bar. The solid line is a predicted curve using 
Eq. 6.9 which includes a constant correction. The attenuation due to the walls, Eq. 6.8, 
has been subtracted from the data before fitting. The fit does not include the points below 
80 mK since they are below the signal resolution threshold. 



200 




— n 



P = 5 bars 
f=20MHz 



_L 



Data 
Fit 

i__ 



300 



400 



500 



600 



T(mK) 



Figure 6.21: The natural logarithm of the integrated amplitude vs. temperature at the 
frequency of 20 MHz and a pressure of 5 bars. The solid line is a predicted curve using 
Eq. 6.9 which includes a constant correction. The attenuation due to the walls, Eq. 6.8, 
has been subtracted from the data before fitting. The fit does not include the points below 
50 mK since they are below the signal resolution threshold. 



201 



14.5 


1 ' 
o 


1 > 1 ■ 1 < 1 
Data 


' 1 ' 1 ' 1 ' 1 


1 — 


14.0 




c 


"it 


^s^"^^'*^'^^^ 


- 




— r 


13.5 


- 


^ 


/^ 


_ 


^■^ 


- 


/ 




- 


:3 13.0 


- 


/ 




- 


X 12.5 


- 


/ 




- 


< 


- 


1 




- 


^ 12.0 

c 

11.5 


-J 




\i 




- 


11.0 


- 




y 


f=64MHz 


_ 


10.5 


- t? 
1 


T 


7 

■ /i 1 1 1 1 


P = 1 bar 
I.I 


- 



100 200 300 400 500 600 700 800 900 

T(mK) 



Figure 6.22: The natural logarithm of the integrated amplitude vs. temperature at the 
frequency of 64 MHz and a pressure of 1 bar. The solid line is a predicted curve using 
Eq. 6.9 which includes a constant correction. The attenuation due to the walls, Eq. 6.8, 
has been subtracted from the data before fitting. The fit does not include the points below 
200 mK since they are below the signal resolution threshold. 



202 



14.5 


- 


' 1 ' 1 
o Data 
Fit 


' 1 ' 1 


1 1 1 1 1 


14.0 


— 




5,^<cf^ 


— 


'-^ 13.5 


_ 




/ 


_ 


u 






f 




(0 

X ""S-O 


- 






- 


< 


- 






- 


J 12.5 


- 






- 


12.0 


- 






f=64MHz 


11.5 


- 


1 , 1 / . 1 


1 , 1 


P = 5 bars 

1,1, 



100 200 300 400 500 600 700 

T(mK) 



Figure 6.23: The natural logarithm of the integrated amplitude vs. temperature at the 
frequency of 64 MHz and a pressure of 5 bars. The solid line is a predicted curve using 
Eq. 6.9 which includes a constant correction. The attenuation due to the walls, Eq. 6.8, 
has been subtracted from the data before fitting. The fit does not include the points below 
200 mK since they are below the signal resolution threshold. 



203 

accommodate a continuous sweep of the frequency. Therefore the error due to 
nonparallelism must be estimated. Fortunately, the error due to nonparallelism will occur 
in both the zero and &st sound regimes, and the effect will partially cancel. Nevertheless, 
the sound velocity change between zero and first sound will cause the zeros in the Bessel 
function to appear at slightly different frequencies. Table 6.5 shows the natural logarithm 
of No{a))/N\{(o) expressed as a percent error when compared to the theoretical 
expectation, {h/kgf =5.83 x\0~\mK/MRzf, calculated for the specific value of = 
1x10"^ radians. Table 6.6 shows the percent due to nonparallelism calculated for the 
slightly larger value of = 4 x 10"^ radians. As expected, a relatively small increase in the 
nonparallelism will result in a large increase in the error at high frequencies. In addition, at 
specific frequencies that correspond to a zero in the Bessel function, the error wdll increase 
dramatically. The error at 5 bars is less than the error at 1 bar because there is less 
difference between the first sound and zero sound velocities at that pressure. In the 
previous experiment by Granroth et al. [32], the value of was measured as 
(4.1 ±0.1) X 10"^ radians, which is consistent with errors due to machining. For this 
experiment, the nonparallelism is assumed to be the same as the previous measurement by 
Granroth et al. and the corresponding error is estimated as ± 20% of the final result at 
1 bar and ± 1 0% at 5 bars. The percentage error is large only because the quantum term 
in zero sound attenuation is exceedingly small. 

There are two sources of error that are related to pressure changes in the cell. 
First, there are pressure changes due to variations in temperature. Second, there are 
pressure changes due to the cyclical change (period of 3 days) of the ''He level in the 



204 



64.5 




Time (^s) 



Figure 6.24: The amplitude of a 4 (is received pulse as a function of time and frequency in 
'*He at 1 bar and 30 mK [109]. In the previous experiment by Granroth et al. [32], the 
amplitude of the first echo as a function of frequency was fit to a Bessel function, 
Eq. 6.15, to obtain a value for the non-parallelism of (4.1 + 0.1) x 10^ radians. 



205 



Table 6.5: The possible percent error in the final experimental quantum term using the 
theoretical expectation due to nonparallelism calculated for the specific value of = 
1x10"^ radians. 



Frequency 
(MHz) 


% Error 
Ibar 


% Error 
5 bars 


10 


1.23 


0.480 


20 


1.23 


0.481 


30 


1.24 


0.481 


40 


1.24 


0.482 


50 


1.25 


0.483 


60 


1.26 


0.484 


70 


1.26 


0.486 


80 


1.26 


0.488 


90 


1.27 


0.490 


100 


1.28 


0.493 



Table 6.6: The possible percent error in the final experimental quantum term using the 
theoretical expectation due to nonparallelism calculated for the specific value of = 
4x10 radians. The percent error may be positive or negative. 



Frequency 
(MHz) 


% Error 
Ibar 


% Error 
5 bars 


10 


19.8 


7.71 


20 


20.2 


7.81 


30 


20.9 


7.98 


40 


22.1 


8.23 


50 


23.8 


8.59 


60 


26.5 


9.11 


70 


31.2 


9.85 


80 


40.7 


11.0 


90 


69.1 


12.7 


100 


-33.5 


15.9 



206 

dewar. The pressure change produced by a variation in temperature was measured in a 
subsequent experiment using a strain gauge attached to the ^He cell (see Section 2.9). 
This data indicates approximately linear behavior vdth a slope of -3.16 x 10"^ bars/mK at 
4.72 bars (see Fig. 6.25). As discussed earlier, we have assumed that the volume has 
remained constant and then used the results of Kollar and Vollhardt to compute other 
thermodynamic quantities, including pressure. The pressure calculated in this manner is 
indicated in Fig. 6.25 by the dark triangles. For first sound data, the temperature range 
(100 to 800 mK) corresponds to a AP = -0.1 bars. It is difficult to calculate the effect of 
the pressure change on the transducer properties. However, we can estimate the error by 
remaining at a fixed fi-equency and temperature and changing the pressure. Figure 6.26 is 
the integrated amplitude at 22 MHz and a temperature of approximately 500 mK as a 
fimction of pressure. By combining the pressure change in Fig. 6.25 vs. temperature with 
the amplitude change in Fig. 6.26 vs. pressure, we can estimate the uncertainty in the 
amplitude due to the temperature variation as approximately + 4% of first sound. In 
comparison, the pressure change due to the "^He level in the dewar is significantly less, 
with the error of approximately ± 1% of the first soimd amplitude. 

There is an additional correction due to a change in the acoustic impedance of 
liquid He. The amount of energy that is transmitted into the ^He cell is a fimction of the 
impedance of the transducers and the ^He liquid. The real component of the impedance of 
a hydrodynamic fluid can be written as 

Rc[Z] =p(TJ>)ciT,P), (6.16) 



207 



4.80 



4.75 - 



2 4.70 
(0 

2 4.65 

3 
CO 

w 

- 4.60 



4.55 



4.50 



- 


1 


1 1 1 1 1 1 1 


1 


- 


- 




▲ ^"^CJ 




- 


_ 




A ^^"^"^-^ 




_ 


- 


o 


Data ▲ 


O 


- 


-- 




- Linear fit with a slope of 


A 


- 




- 




AP/AT = -3.16x10^ bars/mK 




- 


- 


▲ 


Pressure change expected from a 




- 


■ 


1 


constant volume of 32.78 cm^ 
1 1 1 1 1 1 1 


1 


- 



100 



200 300 400 

T(mK) 



500 



Figure 6.25: The pressure as measured by a strain gauge instaUed following the 
completion of the zero sound attenuation experiment as a function of temperature. 
Accordingly, a change in pressure will alter the temperature dependence of iirst sound 
attenuation. The solid line is a linear fit. The triangles represent the pressure change 
expected with a constant volume of 32.78 cm^ [97] 



3 
(0 



240 



238 



236 



234 



232 



230 



^ I ' 
• Data 

Linear Fit 



-1 L 



f=22 IVIHz 
T- 500 mK 



4.65 4.70 



4.75 4.80 4.85 

Pressure (bars) 



4.90 



Figure 6.26: The integrated amplitude of a 22 MHz pulse as a function of pressure at a 
temperature of approximately 500 mK. 



208 

where p(T,P) is the density of the liquid and c(T,P) is the sound velocity. Between zero 
sound and &st sound, there is a signijficant increase in the sound velocity and a subtle 
change in p, combining to cause an increase in acoustic impedance. Independent of 
frequency, the overall signal amplitude will decrease in the first sound regime. 
Accordingly, a constant adjustment must be applied to the first sound data to force a zero 
intercept in a linear fit of the data in Figs. 6.27, 6.28, and 6.30. The constant adjustment is 
0.466 cm"' and 0.537 cm"' for the 1 and 5 bars data, respectively. The ratio between 
these two values (0.537/0.436) = 1.15 is roughly equal to the ratio between the calculated 
values ofp(5 bars)c(5 hars)/p(\ bar)c(l bar) « 1.38 [61]. 

The data in the first sound regime is fit to the temperature dependence in Eq. 6.10 
to derive the frequency dependent constant F(co). This constant is the zero attenuation 
amplitude of the transducers at each frequency. By comparing the integrated amplitude of 
the received zero sound signals with this constant value, we can determine the absolute 
attenuation in zero sound, Ocor. The final result is a plot of {a,„, /a'(P)T^Y^ -l} vs. 

{o)/2;ry, shown in Fig. 6.27 (P = 1 bar) and Fig. 6.28 (P - 5 bars). According to 
Landau's prediction, the data are expected to fall on a straight line with a slope of 

(^/^s)^ =5.83xl0~^(mK/MHz)^ The experimental result at a pressure of 1 bar 
indicates that the best fit line through the data is larger than Landau's prediction 
(2.6 ± 0.5). The error bars encompass all of the theoretical and experimental uncertainties 
including the non-parallelism, and uncertainty in the pressure, value of a'{P), and the path 
length, /. For 5 bars, the slope is slightly negative which means that we are not taking into 
account a source of attenuation in the first sound regime at the pressure of 5 bars. The 



209 

quantity that is most uncertain at 5 bars is the second term in the longitudinal viscosity, r2 
(see Eq. 6.1 1), and this may be responsible for the discrepancy, as well as the relatively 
poor fit of the data shown in Fig. 6.23. Although we could assume a form for the 
viscosity at 1 and 5 bars that would explain the amplitude vs. temperature dependence of 
first sound as well as match Landau's prediction for zero sound attenuation, presently 
there is no independent way of determining both the second viscosity and the quantum 
term. 

In conclusion, the results at the pressure of 1 bar are larger than the predicted 
quantum term in zero sound, i.e. {[«,„, /a'(P)T^]-\}T^ = (2.6 ± 0.5){ho}/2;rksf . With 

the second term in the longitudinal viscosity extrapolated linearly fi-om saturated vapor 
pressure, the overall result at 5 bars is a null result (see Fig. 6.28). A similar result was 
obtained by Matsumoto et al. [31], who measured the predicted quantum term at 1 bar 
and a null result at 5 bars. Perhaps, this result is produced by the unknown contribution 
fi-om the second viscosity, which may become important at higher pressure. An additional 
experimental consideration is that the thermal conductivity is an important contribution to 
the first sound attenuation due to its o) dependence. 



210 



~~l ' 1 ' 1 ' 1 '- 

Present work at 1 bar (1999) 

using a viscosity of ^ = 1 .99/T^ + 4.44x1 0''/T°* 
■Landau's prediction, m = {tiJkgf 




_L 



_L 



2000 



4000 



6000 



8000 10000 12000 



Frequency^ (MHz^) 



Figure 6.27: The corrected and normalized attenuation {a^^^ /a'iP)T^y^ -l} plotted as 
a function of {collnY . The triangles indicate the results at 1 bar using a viscosity of 
7= 1.99/T^ + AAAx\Q^rf^^ and the solid line represents the prediction of Landau. 



Compared to Landau's prediction, the experimental data has 



a slope of 



•exp 



2.6 ±0.5 



211 



1 ' 1 ' 1 ' 1 <- 

Present work at 5 bars (1999) 

using a viscosity of 7 = 1 .62/T^ + 3.62 x lO^H^* 




4000 



6000 



8000 



10000 12000 



Frequency^ (MHz^) 



Figure 6.28: The corrected and normalized attenuation Ja^^^ /a'iP)T^y^ -l} plotted as 
a fimction of {o)/2;ry . The triangles indicate the results at 5 bars using a viscosity of 
1.62/T + 3.62 X IO^'/T"''^ and the solid line represents the prediction of Landau. 



7 



Compared to Landau's prediction, the experimental data has a slope that is slightly 



negative, which is unphysical, and hence we arrive at a slope of '"^'^ 



h^ksY} 



= 0.0±1.0. 



CHAPTER 7 

HE ENERGY 

THE LOW TEMPERATURE LIMIT 



DIRECT MEASUREMENT OF THE ENERGY GAP OF SUPERFLUID ^HE-B IN 



The pairing energy, 2A(7), of the Cooper pairs in superconductors and superfluids 
is a fiindamental characteristic of these systems as all their properties are linked to this 
scale. In fact, since most of the experimentally accessible quantities are expressed in 
terms of the value of the energy gap at zero temperature, A(0), accurate measurements of 
this energy scale are desirable. In the limit of weak-coupling, BCS theory provided the 
first theoretical estimate of this important energy, namely Abcs(0)/Ab7c ^ 1.76 [34] and 
this result is also valid for superfluid ^He in this limit [33,97]. Furthermore, A(7) is 
expected to approach its fiilly developed, low temperature value when 777c < 0.25, where 
7c is the transition temperature fi"om the normal to the superconducting state. However, 
in real systems, the ideal weak coupling limit is rarely realized, and this situation clearly 
arises in the case of superfluid ^He whose phase diagram in zero magnetic field 
demonstrates the presence of strong-coupling corrections to the pairing mechanisms [97]. 
Nevertheless, in zero magnetic field, superfluid ^He-B is expected to possess an isotropic 
energy gap whose gross characteristics are described by BCS theory. Deviations fi"om the 
weak-coupling limit have been modeled by Serene and Rainer [35] who used quasi- 
classical techniques to incorporate strong-coupling corrections in the weak-coupling plus 
(WCP) model. The resulting energy gap, A+(P,7), may be calculated by using 7c(P) and 
6C/Ca<P), the jump in the specific heat at Tc, as input parameters [35,61]. 

212 



213 



The measurement of the energy gap in superconductors can be made in several 
ways, including tunneling and far-infrared spectroscopy. The low-temperature limit, i.e. 
T/Tc < 0.25, is easily accessible using current dilution refrigerator technology. For the 
case of superfluid ^He, tunneling is impractical. However, ultrasonic techniques are well 
suited to study 2A(P,7), as a well defined zero sound mode propagates [61]. Since the 
superfluid transition temperature 7c(P), varies from approximately 1 to 3 mK, depending 
on the pressure, the pairing energy is accessible by radio frequency (50-225 MHz) 
spectroscopy. In fact, when the ultrasonic frequency exceeds 2A{P,T)/h, a sudden 
increase in attenuation is observed as pair breaking occurs. 

Although several studies were made close to Tc [61], one of the first experiments 
to measure 2A{P,T < Tc) was reported by Adenwalla et al. [36], who worked at 
TITc > 0.6 and between 2 and 28 bars. These authors concluded that their measurements 
were consistent with the predictions of the WCP model. At about the same time, 
Movshovich, Kim, and Lee [37], working in finite magnetic fields (0.9 kG < H < 4.6 kG) 
and over a range of pressures (6.0 bars < P < 29.6 bars) and temperatures (0.3 < T/Tc < 
0.5), interpolated their data to the zero temperature and zero magnetic field limit to 
extract values for A{P,T -> 0) at 4.8, 9.8, and 18.1 bars. These authors concluded that 
their values above 9.8 bars were comparable with the WCP predictions, whereas the 
energy gap at 4.8 bars was approximately equal to the BCS value. Since the work of 
Movshovich, Kim, and Lee, no systematic and direct attempt has been made to measure 
the gap in the low temperature and low pressure limit. In fact, the results of Movshovich, 
Kim, and Lee have been largely ignored, as most researchers studying superfluid ^He-B at 



214 



low pressure use A+(P,T^. For example, an ultralow temperature scale, which uses 
A+(P,7) for calibration purposes, has been proposed [1 10]. 

The conventional method of using ultrasound involves sweeping the temperature 
while operating at an odd harmonic of a high-Q transducer. This method is suitable above 
777c > 0.5, where A{P,T) increases significantly as the temperature is decreased. 
However, below T/Tc < 0.3, A(P,7) starts to become temperature independent, so 
alternative methods must be employed. One possibility is to vary isothermally the pressure 
or magnetic field in order to tune 2A{P,T)/h to the operating fi-equency of the transducer. 
This approach was used by Movshovich, Kim, and Lee, who varied the pressure in 
different magnetic fields in order to identify a kink in the response of their acoustic signal 
[37]. This kink was interpreted as the 2A pair breaking edge, and these data were then 
extrapolated back to zero magnetic field limit to extract a value for A(P,T -> 0). A 
broadband frequency approach to the experiments at low temperature would be more 
favorable, and there have been several attempts using plastic (PVDF) transducers 
[111,112]. However, these transducers were unable to operate above 60 MHz and the 
2A/h edge was not observed. 

In this chapter, I report the direct measurement of the 2A pair breaking energy in 
superfluid He-B as a function of pressure and temperature. These measurements have 
been carried out using pulsed FT ultrasonic spectroscopy. The general details of this 
experiment have been described in Section 2.9 of this dissertation. The measurements 
were performed at sufficiently low temperature so that the results are independent of the 
choice of thermometry or temperature scale. The experiment used commerciaUy available 



215 



LiNbOs transducers with frequency windows of approximately 4 MHz centered on the 
odd harmonics of a 21 MHz fundamental frequency. The two frequency windows that 
correspond to the transducer 3"* and S"' harmonics are at a high enough frequency (v > 
A(0)//j) and have sufficient response to observe the 2A energy gap. The use of wideband 
transducers allows direct observation of the 2A//j frequency edge as well as greater 
flexibility in sweeping the temperature or pressure. Since no external magnetic field was 
applied to the sample, the order parameter was not distorted as it was in the study by 
Movshovich, Kim, and Lee [37]. The edge frequency is determined directly from the FT 
of the received pulse, and no additional extrapolation is required. 

This chapter has been divided into seven sections which individually present a 
particular aspect of the experiment. The first section discusses experimental details 
specific to the 2A acoustic measurement. In the second section, thermometry issues 
relating to the thermal connection between the nuclear stage temperature and the sample 
are presented. The third section discusses the possible decrease of the edge frequency due 
to the depletion of the Cooper pair density by the input pulse. The fourth and fifth 
sections present the temperature and pressure dependence of the 2A edge, respectively, 
and also make comparisons to theoretical predictions. The sixth section deals with 
possible sources of error in the identification of the edge frequency from the FT data. The 
final section discusses the calibration of the absolute attenuation above the 2Mh edge 
frequency using the measured zero sound attenuation above Tc. 



216 



7.1 Details of the FT Spectroscopy Technique 
A typical pulse sequence consisted of 128 pulses with 4 s delay between each 
pulse. A phase cycling procedure was used for averaging over multiples of 4 pulses. A 
short, 0.4 (IS, pulse was chosen to increase the pulse bandwidth and also to decrease the 
energy input into the ^He liquid. The output level of the spectrometer was 1 3 dBm and 
the signals were attenuated by approximately -30 dB before reaching the cell so that 
during a data acquisition, the pulse power into the liquid was typically 1 nJ. The returning 
signals were amplified by approximately 20 dB before reaching the spectrometer. The in- 
phase and quadrature components were separately digitized at 10 M sample/s (100 ns time 
resolution) for 2048 samples. The spectrometer would cycle through as many as eight 
different fi^equencies during a single experiment. At each fi-equency, a Pt NMR trace 
would be recorded before and after data collection. The integrated value of each trace 
was averaged to determine the temperature. There was at least 8 minutes between data 
acquisition periods. Care was taken to insure that the signals remained in the linear 
response regime [113]. 

The amplitude of a typical received pulse vs. time using an excitation frequency of 
107.0 MHz at 250 ^K and 4.72 bars is shown in Fig. 7.1. The direct pulse and the echo 
have been indicated using the black and dark gray lines, respectively. The first few 
microseconds of data contain coherent noise related to the spectrometer internal switching 
mechanisms. This noise resembles a large amplitude pulse approximately 1 i^s wide. If 
this signal is not blanked, the FT signal of the received pulse will be the convolution of the 
actual signal and the spurious 1 \xs pulse. The region after the direct signal can also be set 



217 



to zero to remove additional structure in the FT related to the periodicity of the signal 
produced by the echo. This additional structure is related to the cell size and is not a 
property of the liquid. Although blanking portions of the signal to zero will improve the 
FT data, we need to investigate the possibility that this blanking procedure may introduce 
structure in the FT that did not previously exist or may change the position of the 2A 
edge. To this end, we compare the FTs of the previous signal. Fig. 7.1, blanked in 
different ways. Figure 7.2 shows the amplitude of the power spectra for the entire signal, 
the direct signal (echo blanked), and the echo (direct signal blanked). Additionally, for all 
three of the FTs shown in Figure 7.2, the iirst 15 (as of data are blanked to zero for the 
reasons mentioned above. The frequency cut-off edge is clearly evident in Fig. 7.2 as a 
sudden increase in attenuation. We interpret this feature as arising from the 2A{P,T) pair 
breaking phenomena. The blanking procedure has not altered the edge frequency in any of 
the three cases. Furthermore, blanking has reduced the noise level above the edge 
frequency. The highly structure response is typical of the received pulses and is related to 
properties of the transducers [39]. Although the bandwidth is structured as a fimction of 
the operating/response frequency, this limitation is not relevant for the present experiment 
which simply seeks to identify the crossover from low to high attenuation. The response 
fimctions and frequency response of the LiNbOa transducers have been discussed in 
Section 2.9. For the remainder of the data reported in this chapter, the first 15 ^is of data 
and the echo were blanked to zero before taking the FT. 



218 



1.00 



0.75 

■S 0.50 
Q. 

E 
< 

0.25 



0.00 



-1 — 1 — I — r 



n 1 1 T" 



50 



1 1 1 1 1 1 r- 

Direct Transmission Only 

Echo Only 



v^= 107.0 MHz 
7=250nK 
P= 4.72 bars 




100 

Time (usee) 



ISO 



200 



Figure 7. 1 : The amplitude of the received signal vs. time at 7 = 250 \iK and P = 
A. 12 bars. The black line indicates the region designated as the received pulse and the 
dark gray line indicates the region designated as the echo. 



0,6 



0) 
T3 

E 
< 



0.4 



0.2 



0.0 



FFT of Total Signal 
- FFT of Direct Signal Only 
• FFT of Echo Only 



v^= 107.0 MHz 
r=250nK 
P= 4.72 bars 




.1 1 .. iii , J. i . 



105 



106 107 

Frequency (MHz) 



108 



Figure 7.2: The amplitude of the power spectra of the received pulse, indicated in 
Fig. 7.1, clearly showing the IdJh frequency cutoff. For the direct signal and echo, the 
remaining regions of the signal have been blanked to zero. 



219 



7.2 Thermometry Issues 
Throughout this chapter, the reported temperatures were measured using a Pt 
NMR thermometer which was calibrated above Tn with a ^He melting curve thermometer. 
Details of this calibration are discussed in Section 2.9. During the experiment, nuclear 
stage temperatures of about 100 |aK were routinely reached. In the T -> limit, the BCS 
and WCP theories predict that the 2A//7 edge frequency should approach a constant value, 
2A(0)//7. Therefore, it is important to know the lowest actual liquid temperature so we 
can estimate our possible error in the frequency measurement. Figure 7.3 shows the 
integrated amplitude of the received pulse vs. T/Tc at 0.14 bars and 64 MHz, where the 
integration range was limited to the 0.4 ^s received pulse. The lowest temperature, as 
indicated by the Pt NMR thermometer, was 120 ^iK. The vertical line at T/Tc = 0.23 (7 = 
220 ^K) indicates the temperature at which the ^He sample becomes thermally 
disconnected from the nuclear stage temperature as measured by the Pt NMR 
thermometer. The increase in the integrated amplitude of the received signal as the 
temperature is lowered is due in part to the movement of the edge frequency as well as a 
decrease in zero sound attenuation. In other words, the signal increase is due partially to 
the fact that at lower temperatures, the 2A//7 edge is at a higher frequency and hence there 
are more frequency components in the received pulse. However, analysis performed using 
the integrated amplitude of only the FT spectra frequencies below the edge frequency, 
gives similar results. The horizontal line indicates the signal level below which it is 
impossible to distinguish the returned signal from the noise. Consequently, typical 
measurement of the temperature dependence of ^He-B was limited to the range from 



220 



3 

Q> 
TJ 

3 

"5. 

E 

< 

(D 

u 
Q. 

•D 

0) 

-*— > 

CO 
0) 



30 
25 


' 1 ' 1 ' 1 ' 1 

8 o ^ 

O 


1 ' 1 ' 
P= 0.14 bars " 
V =64 MHz - 

ex 




Thermal 


o 


■ 


20 


Disconnect 




— 




- 


o 


■ 


15 


- 


o 
o 


- 




- 




■ 


10 


— 


o 


_ 




„ 


o 


_ 


5 


- Noise Level 


o 


_ 








° n n r^^r\^ 0„ 


n 


. 1 , 1 


1.1,1,1, 



0.0 



0.1 



0.2 



0.3 0.4 

T/T. 



0.5 



0.6 



0.7 



Figure 7.3: The amplitude integrated over the 0.4 [is window of the received pulse vs. 
T/Tc using an excitation frequency of 64 MHz at 0.14 bars. The vertical line at T/Tc = 
0.22 indicates the temperature at which the ^He cell becomes thermally disconnected from 
the nuclear stage temperature as measured by the Pt NMR thermometer. The horizontal 
line indicates the signal level below which it is impossible to distinguish the returned signal 
from the noise. 



221 



0.2 < TITc < 0.6. Similar determinations of the lowest liquid temperature using the 
integrated amplitude of the pulse were performed for each pressure. Figure 7.4 shows the 
amplitude integrated over the 0.4 |j,s window of the received pulse vs. T/Tc using an 
excitation frequency of 107 MHz at 4.7 bars. Again, the vertical line at T/Tc = 0.22 (T = 
320 |iK) indicates the temperature at which the ^He cell becomes thermally disconnected 
from the nuclear stage temperature as measured by the Pt NMR thermometer. Ergo, 
TmJTc = 0.22 was pressure independent for the two pressure regions when performing a 
fiill pre-cool of the nuclear stage to approximately 7 mK. 



7.3 Edge Effects 
To ensure that the observed cut-off features represent the 2A{P,T) pair-breaking 
phenomena, data were also acquired under isothermal and isobaric conditions while 
sweeping the excitation frequency, Vex. The amplitude of the FT power spectra of the 
received signals at T/Tc = 0.27 and P = 0.15 bars is shovm for five different excitation 
frequencies, Vex, in Fig. 7.5 [38]. The temperature and pressure changes were less than 
50 i^K and 0.02 bars, respectively, during this measurement. For all of the traces shown, 
the first 15 us and the echo were zero blanked using the procedure described in the 
previous section. The edge frequency is plainly evident and appears to be the same 
location for all excitation frequencies. If a significant quantity of quasiparticles are excited 
by the pair-breaking process, then the energy gap should decrease, and this possible 
distortion may be tested by sweeping Vex. When Vex increases, additional pair-breaking 



222 



u 


1 1 i III 


1 ■ 1 ' 1 ' f 


d 
^ 


. 


P = 4.7 bars 


g^oocj 


v^= 107.0 MHz 


^ 5 


ex 


=J 


1 o 


^ 


H— ' 




O ^ 


< 4 


Thermal i 


■ 


Disconnect i 


O 


0) 






<n 


1 


o 


12 






CL ^ 






3 


^ 1 




■D '^ 




^ 


0) 






4— • 






(D 


' 


o 


i_ 






D) 






il> 2 


_ Noise Level ' 


o 


^ 




nt/^ ^^ ^ 0\0 n r\ 




^ — 

1 . 1 ' . 


1 • 1 . 1 . 1 


0. 


0.1 0.2 


0.3 0.4 0.5 0.6 

T/T. 



Figure 7.4: The amplitude integrated over the 0.4 ps window of the received pulse vs. 
T/Tc using an excitation frequency of 107 MHz at 4.7 bars. The vertical line at T/Tc = 
0.22 indicates the temperature at which the ^He sample becomes thermally disconnected 
from the nuclear stage temperature as measured by the Pt NMR thermometer. The 
horizontal line indicates the signal level below which it is impossible to distinguish the 
returned signal from the noise. 



223 



T/r =0.27 






V =68.0 MHz 



rJlMMJLM.j\fvAX^VA.«to^.Jw.JL»w.^^>^ 



V =67.6 MHz 

ex 




65 66 67 68 

Frequency (MHz) 



70 



Figure 7.5: Amplitude of the FT power spectra of the received signals at 777c = 0.27 and 
P = 0.15 bars is shown for five different excitation fi-equencies, Vex. The edge fi-equency is 
independent of the excitation fi^equencies. 



224 



occurs because of the increase of additional Fourier components with frequencies greater 
than 2A(P,7)//7. In Fig. 7.5, the sharp cut-off edge is independent of Vex, thereby 
suggesting that our determination is not influenced by excitation dependent effects. It is 
noteworthy that some finite transmission is present above the cut-off edge when Vex is 
greater than 2A{P,T)/h. Although the origin of this effect has not been determined, we 
have considered the small dispersion distortion of the pair breaking edge [114]. 

Knowing the input power of the pulse, we are able to estimate the possible 
frequency shift due to the depletion of the Cooper pair density. From the dimensions of 
the cylindrical sample cell {R = 0.375 cm and / = 0.322 cm) and the density at this 
pressure, the number of moles of ^He is easily calculated (A'moi = 3.86 x 10~^ mol). The 
number of Cooper pairs can then be estimated roughly by considering the energy needed 
to break each pair, 

"pa.. s(A^^,/2)[l-exp(-2A/^,r)]s(A^^,/2) = 1.16xl0^' pairs. (7.1) 

In short, the numbers of Cooper pairs is virtually equal to half the number of 'He atoms at 
such a low temperature. By estimating that the number of quasiparticles formed by each 
pulse is proportional to the energy per pulse, 



E « 1 nJ / pulse - n^ (A/"/ 67.25 MHzX2A), (7.2) 



225 



we can determine the shift in fi-equency as A/ ~ 2.5 kHz. Figure 7.6 shows an expanded 
view of the 2 A edge shown in Fig. 7.5. On this scale, possibly a small softening of the 
edge is evident as the excitation frequency is increased. The temperature is constant 
within the uncertainty of the thermometer and the pressure is decreasing slightly from top 
to bottom (~ 0.005 bars), which should cause the edge frequency to decrease (~ 40 kHz) 
from top to bottom. However, it is impossible to determine if such a small frequency shift 
is occurring since the frequency resolution is 5 kHz. In conclusion, we note that a small 
decrease in the 2A edge is possible when the excitation frequency is higher than the edge 
frequency. This frequency shift A/~ 2.5 kHz can be understood as the depletion of the 
density of Cooper pairs by the pulse input power. The absorbed power increases as the 
excitation frequency is increased above the 2A/^ edge, i.e. more Fourier components are 
attenuated. A frequency shift of this magnitude is too small to be detected using our FT 
technique and does not affect the error analysis. 

7.4 Temperature Dependence 
A set of typical data is shown in Fig. 7.7, where the amplitude of the FT response 
is shown as a fimction of T/Tc at 4.7 bars when the excitation frequency Vex = 106 MHz. 
For T/Tc < 0.3, the detected spectra are clearly cut-off at high frequencies, which defines 
the transition from the low to high attenuation regime. At higher temperatures in the 
superfluid, the cut-off edge moves to lower frequencies and is less sharp. Furthermore, 
the entire FT response is attenuated when compared to the results at low T/Tc. Although 
not completely obvious from the presentation in Fig. 7.7, the softening of the cut-oflf 



226 



3 

CO 

3 




66.4 66.6 66.8 67.0 67.2 67.4 67.6 67.8 68.0 

Frequency (MHz) 



Figure 7.6: An expanded view of the amplitude of the FT power spectra of the received 
signals at T/Tc = Q21 and P = 0.15 bars is shown for five different excitation fi-equencies. 
Vex. From the top to bottom trace, the measured temperature is constant within 1 |nK and 
the pressure is decreasing slightly (~ 0.005 bars), which should move sUghtly lower 
(~ 40 kHz). It is impossible to determine if a fi-equency shift is occurring due to Cooper 
pair depletion since the expected size of the effect (A/~ 2 kHz) is smaUer than the 5kHz 
resolution of the FT. 



227 




106 107 108 

Frequency (MHz) 



Figure 7.7: The FT power spectra of the received signals are shown as a function of T/Tc 
when Vex =106 MHz and P = 4.7 bars. The structure of the spectra is an artifact of ow 
technique [39], but the 2 A cut-off edge is clearly visible. The frequency cut-oflF predicted 
by WCP 2A+(P = 4.7 bars, 7) [61] is given by the broken line. 



228 



feature and the increased amount of response that is detected above 2A(P,7) arises from 
an increase in the population of thermally activated quasiparticles, whose density would 
vary as «exp(-A/r). A similar result is obtained at the lower pressure window of 
0.14 bars, see Fig. 7.8. 

The broken line in Figs. 7.7 and 7.8 is the WCP prediction of the llsjh edge 
frequency as a fiinction of T/Tc. Although the magnitude of the WCP prediction is much 
higher that our experimental edge, the temperature dependence appears to be consistent. 
Figure 7.9 shows a typical graph of the 2Mh edge frequency vs. T/Tc at 4.72 bars with the 
frequencies corrected for pressure drift. The upper and lower triangles correspond to the 
positive and negative error in the frequency determination. When multiplied by a constant 
factor, the WCP and BCS theoretical curves agree, within error, with the data. The solid 
line is the WCP prediction multiplied by the factor 0.981 ± 0.001 and the dashed line is the 
BCS prediction multiplied by the factor 1.003 ± 0.001. For the lower pressure window of 
0.15 bars, these factors can be determined in a similar fashion as 0.955 + 0.001 for WCP 
and 0.961 ± 0.001 for BCS. Above a TITc of 0.3, the uncertainty in the determination of 
the edge frequency increases dramatically and it is impossible to determine the exact form 
of the temperature dependence. The temperature dependence of the edge agrees, within a 
constant factor, with the fimctional form of either the BCS or WCP model, but a 
somewhat better agreement is given by the WCP predictions. 



229 



0.1 




0.4 



0.5 



juJi ■*"llll|,||-^ ■' I 



'*'■* ■"-■^'■■--" ■ 



iHjJJiiadJtwti.t-iM 



lilJillUU 



mm 



'^"•'-ftdh ' ■- -■ 



^^■^" ■" ■ -^ 



i.IllI. Jiuijtt.^. ... 



M, i nt>>*>lt l l*iiiJ n >li»ti»iL. n it „ i 



.Mkl 



i i t>i<iui< n <iiiUikirt«M u lijioi 



■'^ iii-.^...iL — ■■ 



2A 



V =66.8 MHz 

ex 

P = 0.14 bans 



63 64 65 66 67 68 

Frequency (MHz) 



69 



70 



71 



Figure 7.8: The FT power spectra of the received signals are shown as a function of T/Tc 
when Vex = 66.8 MHz and P = 0.14 bars. The structure of the spectra is an artifact of our 
technique [39], but the 2A cut-oflFedge is clearly visible. The frequency cut-off predicted 
by WCP 2A+(P = 0.14 bars, 7) [61] is given by the broken line. It should be noted that the 
excitation frequency is higher than the 2A/h edge frequency which produces the response 
above the edge (see Fig. 7.5). 



230 





lUO.U 


- 


V 


1 ' I ' ' ' 1 ' ' ' 

Highest Frequency Identification of 2a(T)//j 






. 


A 


Lowest Frequency Identification of 2A{T)/h 


I 


107.5 
107.0 
106.5 


- 




-BCS * 1.003 
- WCP * 0.980 


— 


c 
0) 

LL 


■ 




~«>~^*~ -<r=5=^=^=>^ 


^s^ Y Y 






_ 


^Px = 


= 107.0 MHz 






inR n 


- 


p = 

1 


4.72 bars 
1 1 1 1 ■ 


\ ■ 



0.1 



0.2 



0.3 



0.4 



T/T, 



Figure 7.9: The 2A edge frequency as a fianction of T/Tc using an excitation frequency of 
107.0 MHz at 4.72 bars. The upper and lower triangles indicate the possible error in the 
edge frequency. The solid and dashed lines correspond to BCS and WCP theory 
multiplied by a constant factor for comparison with the data. 



231 



7.5 Pressure Dependence 

To insure that the observed cut-oif features represent the 2A{P,T) pair-breaking 
phenomena, data were also acquired under isothermal conditions while varying the 
pressure with fixed v^x. Figure 7.10 shows the amplitude of the FT power spectra of the 
received pulses at a temperature of approximately 250 |j.K for five closely spaced 
pressures fi-om 4.74 to 4.80 bars. In Fig. 7.10, it is observed that the fi^equency of the 2A 
edge increases as the pressure increases. However, the fine structure of the FT away fi"om 
the 2A edge remains unchanged. Consequently, the fine structures in the response cannot 
be associated with any collective modes, which should vary with pressure [61,115]. The 
fine structure in the FT power spectra has been discussed previously and is a consequence 
of the properties of the transducers and our technique [39]. 

The dispersion of superfluid ^He-B is characterized by order parameter collective 
modes which are specified by their total angular momentum, J. The J = +2 and J - -2 
modes are produced fi-om p-wave pairing interactions and are termed the real and 
imaginary squashing modes, respectively. Higher order interactions (J > 2) do not 
contribute to the equilibrium properties of ^He in zero magnetic field [116]. However, 
Sauls and Serene [115] have predicted that the J = 4- mode, will be observable in 
ultrasonic attenuation experiments. This mode, produced by f-wave pairing interactions, 
should be discernible as a sharp resonance just below the 2A/h fi-equency edge [116]. 
Although higher order interactions cannot alter the 2A/h fi-equency directly, we must 
explore the possibility that the attenuation edge, which we have attributed to 2A pair 
breaking may be produced by the J = 4- mode or a combination of the 2A/h edge and 



232 



3 




106 107 

Frequency (MHz) 



108 



Figure 7.10: Amplitude of the FT power spectra of the received signals shown as a 
function of pressure when Tw 250 ^K and Vex =106.8 MHz. 



233 



J= 4- resonance. If the J = 4- mode is weakly coupled to the zero sound excitation, the 
resonance will be sharp, and we should expect to observe a recovered signal at higher 
frequencies above the J = 4- resonance but before the INh edge. Therefore, the J = 4- 
mode width would have to span ~ 4 MHz, which is unlikely if it weakly coupled to the 
probing sound wave. We can rule out this possibility by noting that in experiments where 
the center frequency was above the 2A//2 edge, a recovered signal was never observed (see 
Fig. 7.11 (A)). On the other hand, if the J = 4- mode is strongly coupled to the zero 
sound excitation, then we should expect to observe significant dispersion effects and 
possibly a recovered signal at higher frequencies (see Fig. 7.1 1 (B)). These effects would 
be apparent as a frequency dependent group velocity. The received pulses at two different 
frequencies shown in Fig. 7.12 demonstrate that the time-of-flight of the initial received 
pulse does not depend on the excitation frequency. However, in this experiment, only 
0.4 jj,s pulses were used and the frequency bandwidth of a single pulse is approximately 
2.5 MHz. Consequently, it is difiScult to rule out the possibility of dispersion effects, from 
this argument alone, as the frequency difference is almost the same magnitude as the 
bandwidth. However, at low temperatures, we always observed a sharp edge in the 
frequency spectrum corresponding to the 2A edge with no apparent curvature resulting 
from dispersion effects. 

All of our data in the low temperature limit, i.e. T/Tc < 0.25, may be combined in 
one plot which compares our results with the predictions of the WCP and BCS models. 
The theoretical and experimental values are plotted as a fimction of pressure. Fig. 7.13, 
and density. Fig. 7.14. It is important to stress that the experimental data in Fig. 7.13 and 



234 



Fig. 7.14 do not depend on thermometry or our choice of temperature scale. On the other 
hand, the theoretical predictions require inputting TdP) and, in the case of the WCP 
model, 5C/Ca<P). Finally, our values for the energy gap in the low temperature liquid may 
be normalized by ksTdP) and may be compared to the results of Movshovich, Kim, and 
Lee [37] and theoretical predictions. Fig. 7.15, where a choice of TdP) and bC/C^fP) 
have to be made. This same data is plotted as a function of density in Fig. 7.16. For the 
purpose of consistent comparison in Figs. 7.13, 7.14, 7.15 and 7.16, the parameters 
compiled by Halperin and Varoquaux [61] have been used. From extrapolation, it 
appears that the 4.8 and 9.8 bars edge measurements of Movshovich, Kim, and Lee are 
consistent with our measurements. The experimental results indicate that A(P, T ^ 0) is 
smaller than theoretically expected, even if one allows for an uncertainty of TdP) of 
approximately 1%. The most plausible explanation for the difference is that the 
experimentally determined value of Tc is in error. The 1% uncertainty is an estimate based 
on the history of the superfluid phase diagram, i.e. TdP). However, it should be noted 
that the value of the antiferromagnetic transition for solid ^He at the melting curve, i.e. Tn, 
is estimated to be 0.94 mK ± 3% [117]. Consequently, a 3% uncertainty of T^ at the 
melting curve may naively be transposed to the liquid at low pressure where TdP ~ SVP) 
= 0.93 mK ± 3%. Nevertheless, this allowance does not completely resolve the 
discrepancy of 4.5% that has been established by our measurements. A similar trend is 
evident in the energy gap of several conventional superconductors [118], i.e. the energy 
gap is smaUer than that predicted by BCS theory. GeneraUy, the discrepancy is assumed 
to result from an error in the experimentally derived value of Tc. 



235 



1 — ' — I — ' — I — ' — I — I — I — I — I — ^ 




106 107 108 109 110 111 112 113 

Frequency (MHz) 



Figure 7.11: Sketch of the amplitude of the FT power spectrum for a hypothetical 
received pulse. If the edge observed in our FT spectra were produced by a weakly 
coupled collective mode (A), then the we would observe a sharp resonance that would not 
influence our identification of the 2A edge. If the edge observed in our FT spectra were 
produced by a strongly coupled collective mode (B), then we would expect to observe 
significant dispersion of the edge and possibly a recovered signal at higher frequencies. 



236 



16 



T — ' — I — ' — I — ' — I — ' — 4 

66.4 MHz 

68.0 MHz 



T/T^ = 0.27 
P = 0.15 bars 




10 



15 20 25 30 35 40 45 50 

Time (^s) 



Figure 7.12: The received pulse amplitude vs. time for the frequencies of 66.4 MHz and 
68.0 MHz at 0.15 bars and T/Tc ^ 0.27. The sound velocity is frequency independent 
over this frequency range. The purpose of this analysis is to show that any dispersion 
effects are smaller than the measvirement uncertainty, ruling out the possibility that a 
strongly coupled collective mode is responsible for the experimentally observed frequency 
edge. However, in this experiment, only 0.4 )as pulses were used and the frequency 
bandwidth of a single pulse is approximately 2.5 MHz. Consequently, it is difBcult to rule 
out the possibility of a collective mode, from this argument alone, as the frequency 
difference is almost the same magnitude as the bandwidth. 



237 





' 1 ' I ' 1 ' 1 


' 1 


■ 


120 




wcp 










■ 70 

■I 






WCP 


^ 




t= — 




F 




.5 


BCS 




/ 


y^^ . 


110 


"1^68 


_ 




/^i^ 


- 




■ « 






yjiS^ 


■ 




:« 


.,yyVNA/V^ t.^ lOt^. 




^ 


" 




_ LL 


■ 






. 


--V 100 


- 66 


- 


j/"^ j^ 1 


^^.^ 


_ 


N 




1 , , 1 , , 




^ BCS 


. 


^F* 








X 


0.13 0.14 0.15 y^X 






5 




y 










Pressure (bars) ^y^ 


y 1 


' 1 ' 


_ 


- 


>» 


j/^y/^ 


\ 






- 


^ 90 


yy< 


- WCP 




^: 


- 


0) 


/^y^ 




_^ 




■ 


3 


yy ^108 


— 


j^ 


j;^ 


_ 


cr 


// N 




^^■^ 


^y^ 




(U 


yy^ ^ 


- 


.^^ y 


'"'^ 


■ 




yy 5 


M 


^^ 


- 


- 


80 


^<^^^ >, 


^ 


^^ ^^^'^ 


_ 


- 




^/^ 1 107 


'^ 


^\ 


- 


: 




■ X ^ 




BCS 




■ 




y^ Q. 








■ 


70 


A 106 


1 


1 


- 


- 














4.7 


4.8 


4.9- 




\ Pressure (bars) 


; 


fif) 


1 


1 


, 


12 3 4 


5 


6 






Pressun 


3 (bars) 









Figure 7.13: Our 2A(P, T ^ 0) values are shown as a function of pressure. The insets 
show the expanded views of our data near 0.15 and 4.8 bars. The predictions of the WCP 
and BCS models are given by the lines [61]. 



238 



N 

X 



c 

0) 

cr 





7\ 




1 > 1 


' 


1 


1 


120 


■ 70 
■| 69 


J 


WCP ' 




WCP " 




7 


\ 
BCS 




110 


-5^68 


- 






yW"""^ : 




■ S'67 


- 


HfssisH^ ^ ^ 


^s^ - 


y/y^ 




LL 








y\/^ 


100 


_ 66 


- 


1 




v//'^--^ 




. O.OJ 


318 


0.0818 

P (g/cm') 


0.0) 


319 V 


//^ BCS 




. 


y^ 


- / ' 1 ' 1 - 


90 


- 




y/^ 


- WCP ^ - . 




■ 


y 


^ 


80 


■ y 


y 


^108 
S 107 


- X BCS 


70 


/' 




u. 


"X "J 




■ A 




106 


1,1 








0.0914 0.0916 0.0918 ! 


60 


1 


1 . 1 


. . . . P(9/cm^). . ■ 



0.082 



0.084 



0.086 



0.088 



0.090 



0.092 



0.094 



P (g/cm ) 



Figure 7.14: Our 2A(P, T ^ 0) values are shown as a function of density. The insets 
show the expanded views of our data near 0.15 and 4.8 bars. The predictions of the WCP 
and BCS models are given by the lines [61]. 



239 



1.84 - 




A Present work 

V Movshovich et al., (1990) 

Weak-coupling-plus 

BCS 

1 I I I I I I I I I I 1 



Pressure (bars) 



Figure 7.15: The values of 2A(/', T -^ OyksTciP) are shown as a function of pressure. 
The experimental data obtained by Movshovich, Kim, and Lee [37] are displayed along 
with the predictions of the WCP and BCS models [61]. The solid line is a guide to eye. 



240 



1.86 


1 1 1 1 1 1 1 1 1 1— - 1 1 I 1 1 1 1 1 


T 


1.84 
£ 1.82 


wcp "i.-:;:^ 
-^ -••>^ 


- 


o 
5° 1.80 


- /" ' / 


- 


■ — 


-•"■'" / 


. 


o 1.78 


- ...--■""' / ^^BCS 


_ 


t 

1 


^/ r 


- 


•- 1.76 


/ 


- 


q: 


/ V 


- 


^ 1.74 


/ 


- 




/ A Present Work 


- 


1.72 


/ V Movshovichefa/., (1990) 


- 




■ / WeakPlus 


- 


1.70 


- / BCS 


- 


1 Rft 


i_ — 1 — 1 — 1 — i — 1 — 1 — I — 1 — 1 — 1 — 1 — 1 — 1 — 1 — 1 — 1 — 1 — 1 — 


1 



0.08 



0.09 



0.10 



0.11 



0.12 



P (g/cm ) 



Figure 7.16: The values of 2A(P, T -* Q)lkBTc{P) are shown as a function of density. The 
experimental data obtained by Movshovich, Kim, and Lee [37] are displayed along with 
the predictions of the WCP and BCS models [61]. The solid line is a guide to eye. 



241 



In summary, A(P, T -^^ 0) of superfluid ^He-B is not well modeled below 
approximately 10 bars, and our results are significantly below the theoretically predicted 
values. Experimentally underestimating A(P, T ^ 0) is not possible in our measvirements. 
The determination of the edge frequency is not dependent on thermometry issues or the 
choice of the temperature scale. The experiment is performed in zero magnetic field and 
does not require any extrapolation or other calculation to identify the edge frequencies. 
More specifically, magnetic field distortion [37], dispersion eifects [114], corrections to 
our estimates of the minimum 777c, and uncertainties in the pressure of the sample are too 
small to rectify the discrepancy. Consequently, the consistency of our results with the data 
of Movshovich, Kim, and Lee cannot be considered a coincidence. The most probable 
reason for the discrepancy is the uncertainty in the experimentally measured value of 
Tc{P), but the thermodynamic consequences of this conjecture remain to be quantitatively 
verified [97]. 

7.6 Error Analysis 
There are three main sources of error in this experiment: finite temperature, 
distortion of the edge frequency, and uncertainty in the pressure determination. The 
frequency error produced by operating at finite temperature can be easily calculated by 
using a theoretical model, either WCP or BCS, and considering the change in edge 
frequency from T=Oto T= Tmin. At low temperatures, both models approach a constant 
2A{0)/h value. As was discussed in section 7.2, the worst case minimum temperature, 
Tmin, at 4.7 bars was approximately 320 ^iK. Knowing the lowest actual liquid 



242 



temperature, we can determine the possible error. Between T = 320 i^K and T= 0, BCS 
and WCP theory predict a change in frequency of approximately (2A(0)-2A(rMw))//7 = 40 
kHz. A frequency error of 40 kHz corresponds to less than 0.04% error at 107 MHz. 
Clearly, finite temperature is not a significant contribution to the experimental error. 

Second, the worst case A/ produced by the depletion of superfluid pair density by 
the input pulse, when the excitation frequency is above the edge frequency, was estimated 
as less than 3 kHz in Section 7.3. Again, we may safely ignore any contribution from pair 
depletion in the overall error analysis. 

Finally, the largest error contribution is derived from the uncertainty in the 
pressure determination. Hysteresis in the pressure measurement resulted in a maximum 
pressure error of +/- 0.02 bars. Using the slope of the edge frequency vs. pressure in 
Fig. 7.14 (~ 8.12 MHz / bar), we arrive at a frequency error of 160 kHz. Combining all of 
these sources, we can estimate the worst case error in the frequency as 170 kHz. In 
conclusion, the frequency determination is extremely accurate, i.e. considering that the 
lowest excitation frequency in this experiment is approximately 64 MHz, the largest 
possible error is 0.3%. 

7.7 Absolute Attenuation 

It is important to know the absolute value of attenuation in the frequency region 

above the 2A edge, where the attenuation is highest. In order to obtain this value, we 

must compare it to a known attenuation. Figure 7.17 (A) shows the amplitude of the 

received signal integrated over a 0.4 ms window corresponding to the pulse vs. 



243 



temperature at 107 MHz and 4.72 bars. The thermal connection between the liquid and 
the nuclear stage, which was discussed in section 7.2, is clearly visible at approximately 
400 ^iK. The higher minimum temperature {TITc = 0.28 > T^Tc = 0.22) is indicative of 
an incomplete pre-cool of the nuclear stage. The data above Tc have been fit (soUd line in 
Fig. 7.17 (A)) using the following equation for zero sound attenuation, namely 



A = p exp 



a\P)l 



T' + 



( hco ^' 



ylTckgj 



(7.3) 



where A is the ampUtude, / is the path length of 0.322 cm, a\P) is the attenuation 
prefactor, and ^ is a constant related to the initial pulse strength. We know the value of 
the constant prefactor a'{P) = 0.92 cm-'mK"^ fi-om the results of Chapter 6. The 
theoretical zero sound signal (solid line) has been extrapolated to T - to allow a 
comparison with the integrated signal in the superfluid. We can use the data above Tc to 
caUbrated the integrated signal and calculate the attenuation in absolute units (see 
Fig. 7.17 (B)). Using this method, we obtain a coarse estimate of 5.9 ± 0.2 cm"' for the 
maximum attenuation. The velocity of sound determined by the time-of-flight of the 
received pulse is shown in Fig. 7.17 (C). Above Tc, the sound velocity agrees with the 
zero sound velocity at 4.72 bars and deep in the superfluid, T « Tc, the velocity is much 
smaller than either the first or zero sound velocities. We beUeve this result arises fi-om 
dispersion effects below 2A and does not influence our identification of the pair-braking 
edge. 



244 



3 

m^ 

<, 
^ 



E 
o 



(0 

E 



o 
o 

(U 

> 



10 
8 
6 
4 
2 


6 
4 
2 



260 
240 
220 
200 
180 



' ' ' I 



I I I I I I I I I 



I I I I r^ 



T"-^ 




- Noise Level o. 



a©- 



Low Attenuation Threshold 



-O- O -Q 



I I I I I I I I I I I 1 1 I I I I I I I I I I I I I 1 1 I I I I I I 



Noise Level 




o 
o 



p8 ° 



o o 



fiS23333Q33333X) 



v^^= 107.0 MHz - 
P = 4.72 bars 
7c=1-45mK 



'■■■ I ■'■' I '■■■'■'■■'■■■■ I ■■■■ I ■■■■ I 



0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 

T(nnK) 

Figure 7.17: The natural logarithm of the integrated signal vs. temperature using an 
excitation frequency of 107.0 MHz at a pressure of 4.72 bars (A). The solid line indicates 
a fit to the data above Tc to zero sound attenuation, Eq. 7.3. The attenuation due to zero 
sound has been extended below Tc for comparison with the actual attenuation due to pair 
breaking in the superfluid. The dashed lines indicate the noise level and low attenuation 
threshold. The low attenuation threshold is estimated from the Landau Limit experiment 
(Chapter 6) which used a longer 4 ^s pulse. The data from (A) can be displayed as 
attenuation by calibrating against the data above Tc (B). The velocity determined from the 
time-of-flight of the received pulse is shown in (C). In the superfluid phase, the velocity is 
less than both the zero or first sound velocities. We believe this result arises from 
dispersion effects below 2A and does not influence our identification of the pair-braking 
edge. 



CHAPTER 8 
SUMMARY AND FUTURE DIRECTIONS 



This chapter provides a synopsis of the results for the four experiments contained 
in this dissertation. Chapters 4 through 7 are each represented by a section in this chapter. 
In addition to summarizing the achievements of each investigation, possible future 
directions and improvements are proposed. 

8.1BPCB 
8.1.1 Summary 

The crystal structure of the antiferro magnetic insulating compound BPCB has been 
determined in neutron diffraction experiments to consist of isolated two leg ladders. 
Analysis of magnetization measurements (//< 30 T, T> 0.7 K) has allowed us to identify 
the magnetic structure of BPCB as a two-leg, S= \I2 spin ladder that exists in the strong 
coupling limit JJJw -3.5. A single set of exchange constants, Jx = 13.3 K, and 
7|| = 3.8 K, are able to accurately describe all of the data. An unambiguous inflection 
point at half the saturation magnetization, which has been theoretically predicted but never 
observed until now, is well described by an effective XXZ chain [79]. The magnetization 
data near the critical fields Hc\ and Hc2, {T<J\^ exhibit scaling behavior in the universality 
class of the one-dimensional dilute Bose gas transition [2,68]. Although we have 



245 



246 

considered the potential existence of additional exchange interactions, Jf and J', effects 
arising from these parameters are not prominent in the present data. However, since 
subtle differences arise between the theoretical predictions and the data at the lowest 
temperature, additional pertwbing interactions may be present. 

8.1.2 Future Directions 

A ladder model has successfully accounted for the macroscopic magnetic 
properties of BPCB. It would however, be beneficial to confirm the ladder description of 
BPCB with a microscopic probe such as neutron scattering or nuclear magnetic resonance 
(NMR). Initial attempts to directly observe the spin gap in neutron scattering experiments 
have failed due to the small single crystal samples available. In the future, waiting a longer 
time at each wavevector and using a larger single crystal sample or an array of single 
crystal samples should improve the experimental sensitivity. In addition, I propose 
measuring the \IT\ relaxation rate of 'H and ^^Br, using pulsed spin-echo techniques and 
the NMR apparatus described in Chapter 2. Although ^'Br has a quadrapole moment, the 
eight bromine atoms per unit cell are located approximately 2.3 A from the magnetic .S" = 
1/2 Cu^^ sites and appear to be ideally located for investigations of the magnetic exchange. 

8.2 MCCL 
8.2.1 Summary 

The room temperature crystal structure of MCCL indicates that the 5=1/2 copper 
spins are arranged in a chain along the a-axis with an alternating distance between spins. 
Neutron diffraction experiments on single crystals of deuterated MCCL indicate a 



247 

monoclinic to triclinic structural transition in MCCL near 250 K. In addition, there is 
possibly a second structural transition at 50 K. Electron paramagnetic resonance (EPR) 
measurements have been performed on single crystal and powder samples. At all 
temperatures and orientations, the EPR intensity is a single broad line (AH ~ 400 G). The 
structural transition identified in the neutron experiments appears as a discontinuity in the 
EPR linewidth and Lande g factor at 245 ± 5 K. Below 50 K, the Lande g factor remains 
constant and the linewidth increases sharply, suggesting that magnetic exchange is 
producing the transition seen in the neutron diffraction experiments at this temperattire, 
r=50K. 

Low field susceptibility measurements have indicated paramagnetic behavior until 
approximately 10 K, where a deviation occurs, possibly indicating the onset of 
antiferromagnetic order. In addition, fi-om plots of susceptibility multiplied by temperature 
vs. temperature, the structural transition at 245 K can be identified as a sudden change in 
slope. To verify the antiferromagnetic behavior of MCCL below 10 K, high field 
(//< 30 T) low temperature (T > 1.6 K) magnetization studies were performed. At the 
temperatures of 2.0 and 4.3 K, the magnetization curves cannot be explained by a 
Brillouin fiinction. Furthermore, at the lowest temperature of 1 .6 K, the magnetization 
curve is formed by three almost linear regions with distinct slope changes at 3.2 T and 
1 7 T, where the magnetization is saturated above 1 7 T. These slope changes occur at the 
same magnetic fields for both single crystal and powder samples, and the values are not 
sample dependent. Furthermore, there is no evidence for the formation of a spin gap even 
at the lowest temperature of 1 .6 K. Attempts were made to fit the magnetization curves 
using the exact diagonalization method and an alternating chain Hamiltonian, but these 



248 

efforts were unsuccessful. It does, however, appear that the magnetic behavior of MCCL 
may be explained by a combination of free spins and a low dimensional magnetic system. 

8.2.2 Future Directions 

As in the material BPCB, analysis of the macroscopic magnetization results for 
MCCL will be supported by the microscopic information provided by neutron scattering 
and NMR experiments. Further neutron diffraction experiments at low temperature 
(r< 10 K) will help to clarify details of the magnetic exchange. More specifically, the 
magnetic dispersion will indicate if the exchange is antiferromagnetic and if it occurs along 
the chain direction. I also propose NMR measurements of the l/T\ relaxation rate for H 
and '^C, despite the fact that neither atom is directly involved in the superexchange 
between 5=1/2 Cu^^ spins. All of the nuclei in MCCL have an isotope with a net nuclear 
spin. Unfortunately, the nuclei ^^Cl, ^^Cl, '""N, ^^Cu, and ^^Cu have a quadrapole moment 
which complicates the analysis. In addition, the elements Cu and CI have two isotopes 
with a net nuclear spin. Because there are only eight hydrogen and two carbon atoms per 
unit cell, it should be possible to correlate individual lines in the NMR spectra with the 
atomic position [12], as was done for the compoimd Cu(Hp)Cl. Subsequently, the 
relaxation rate, \/T], for each nucleus can be compared to theoretical models [119]. 

8.3 Zero Sound Attenuation in ^He 
8.3.1 Summary 

A low temperature acoustic technique has been used to confirm the quantum term 
in the zero sound attenuation of ^He as predicted by Landau [27]. This second term is 



249 

proportional to the square of frequency and is eflfectively temperature independent. 
Consequently, absolute measurement of the zero sound attenuation is required. The 
amplitude of the received signals in the first sound regime were measured as a fianction of 
temperature and fitted to a known expression to calibrate the zero sound signal amplitude. 
Several discrete frequencies were used for both zero and first sound from 8 to 108 MHz, 
and the experiment was performed at the pressures of 1 and 5 bars. 

The final results at 1 bars are larger than the predicted quantum term, i.e. 
{a,^,/a'iP)T^]r^ -\} = (2.6 ± 0.5) {ha/lTuksf. The results at 5 bars indicate a 

quantum term of zero, i.e. {a,„,/a'iP)T^]r^ -\} = (0.0 ± \.0){ho)/2^kgy . 

Interestingly, a similar, nuU outcome at 5 bars was obtained by Matsumoto et al. [31], 
although no explanation for the result was offered. We have considered two primary 
causes for the measurement of a zero result at high pressure. First, uncertainty in the 
expression for the longitudinal viscosity at 5 bars leads to an overall uncertainty in the 
determination of the quantum term. Next, the second viscosity, which has not previously 
been measured or calculated [100], may become important at higher pressure. 

8.3.2 Future Directions 

Clearly, it is advantageous to measure the absolute zero sound attenuation directly 
and avoid the complications that arise from calibrating using first sound. In this case, 
attenuation contributions from the thermal conductivity as well as the first and second 
viscosities could be ignored, greatly simplifying the analysis. The acoustic path length 
could be changed in an expanding ^He cell through the use of a ''He bellows arrangement. 



250 

The attenuation could then be determined from the measurement of signal amplitude as a 
function of the path length. If calibration of the attenuation using fibrst sound attenuation 
was still required, the longitudinal viscosity should be measured in situ. In addition, it 
may be possible to estimate the magnitude of the second viscosity using the theoretical 
expression by Sykes and Brooker [100]. 

8.4 Measurement of the 2 A Pair Breaking Energy in Superfluid ^He-B 
8.4.1 Summary 

The zero temperature limit of the energy gap A(T,P) of superfluid ^He-B has been 
measured using a novel acoustic Fourier transform technique [38,39] at 777c < 0.25, near 
0.1 and 4.8 bars, and in zero magnetic field. The results of the current experiment show 
that the 2A(T,P) pair-breaking energy is less than predicted by WCP theory at the high 
pressure range {P ~ 4.8 bars) and significantly below both BCS and WCP theory 
predictions at the lower pressure range (P ~ 0. 1 5 bars). The determination of the edge 
frequency is not dependent on thermometry issues or the choice of the temperature scale. 
In addition, the data were acquired in zero magnetic field and extrapolation was not 
required to determine the edge frequency. In summary, A(P, 7 ->^ 0) of superfluid ^He-B 
is not well modeled below approximately 10 bars, and our results are significantly below 
the theoretically predicted values. This conclusion appears to be consistent with the 
results of Movshovich, Kim, and Lee [37]. Experimentally underestimating A(P,T ^ 0) is 
not possible in our measurements. More specifically, magnetic field distortion [37], 
dispersion effects [114], corrections to our estimates of the minimum 7y7c values, and the 
uncertainty in the pressure of the san^)le are too small to rectify the discrepancy. 



251 

8.4.2 Future Directions 

The 2A pair breaking experiment did not depend on the temperature scale or 
external parameters, and the measurement error was extremely small. Nonetheless, the 
measurement of 2A(P) was limited by the available frequency windows that matched the 
value of 2A{P)/h at the pressures of 0.15 and 4.7 bars. A quick method of expanding on 
the current data is to repeat the experiment with a different set of transducers. Preferably, 
the new transducers would have a lower resonant frequency (/^es ~ 10 MHz) and yet still 
have a comparable maximum frequency (/Jnax ~ 100 MHz). The number of frequency 
windows that provide usefiil data should therefore increase. In addition, transducers with 
that are less structured and have improved signal to noise will increase the accuracy of the 
experiment. The sound velocity in the superfluid (Fig. 7.17), near the 2A edge, is 
significantly below both the zero and first sound velocities. This effect may be produced 
by coupling of the sound probe with a collective mode. Measurements of the dispersion 
using longer pulses and the magnetic field dependence will fiirther clarify questions related 
to collective modes located near the 2A edge. 

During the 2A pair breaking energy experiment, care was taken to ensure that 
signals remained in the linear regime. Future experiments may investigate the properties 
of He-B including quasiparticle lifetime and the density of states near the Fermi energy 
using non-linear acoustic methods or "pump-probe techniques". Using similar apparatus 
as the previous experiment, a pulse sequence will consist of an initial pulse or "pump" 
pulse that is designed to deplete the density of quasiparticles at a particular energy and 
then a measurement or "probe" pulse to determine the residual effect after a variable time. 



252 

8.5 Concluding Remarks 

What do low dimensional antiferromagnets and liquid ^He have in common? I 
have struggled to answer this question for approximately 250 pages. Fundamentally, the 
macroscopic properties of both systems require a quantum mechanical description. 
Practically speaking, both topics have been the subject of this dissertation. 

Note added in final proof. After defending this thesis and during the final stage 
of editing, the thesis of Chaboussant, who studied Cu(Hp)Cl, was communicated to 
us [120]. In his thesis, Caboussant provides a plot of dMIdH as a function of H (see 
Fig. 5.11 in Ref 120), which is consistent with our dMIdH plot that was obtained by 
digitizing the published M{H) data (see Fig. 4.19 on page 91). Furthermore, Chaboussant 
compares his MIMs data for Cu(Hp)Cl with two known alternating chain materials (see 
Fig. 5.12 in Ref 120), and from this comparison, he estimates that, within an alternating 
chain description, Ji « 13.4 K and J2 « 4.5 K. These values are in good agreement with 
the ones that we obtained from an exact diagonalization fit of his data when using an 
alternating chain Hamiltonian (see Fig. 4.19 on page 91). However, it is important to note 
Chaboussant believes that, at lower temperatures, the distinctly asymmetric double bumps 
in the dMIdH vs. H plots of Cu(Hp)Cl will evolve into the symmetric double hump 
features expected for a magnetic spin ladder material [120]. After reviewing this 
information, the viewpoint expressed in this thesis remains unchanged, i.e. we believe that 
Cu(Hp)Cl is best classified as an alternating chain system. Undoubtedly, the results of 
inelastic neutron scattering experiments should be able to clarify the classification of the 
magnetic ground state. 



APPENDIX A 
LOW TEMPERATURE PROBE DRAWINGS 



253 



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APPENDIX B 
COIL FORMER DRAWINGS 



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APPENDIX C 
PRESSURE CLAMP DRAWINGS 



277 



278 



Parts 




1/16' 



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rod provided is easy slide fit into 'B' (B~1/4") 



3/a" tap for B-32 

(3 tapped holes) 




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12/B- 




Pressure Clamp (quantity 1) 

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280 



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Pressure Clamp (quantity 1) 

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283 



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BrianWatson, 8133,2-9147 (Melsel) 



284 



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37 jZ. 0*0 ± .006" 




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APPENDIX D 
NMR PROBE DRAWINGS 



286 



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POLYCARBONATE SAMPLE SPACE DRAWINGS 



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APPENDIX F 
APPLESCRIPT ROUTINES FOR DATA ACQUISITION USING THE TECMAG 

SPECTROMETER 

This routine is run from the Applescript editor on a G4 PowerMac. It interfaces with the 
Tecmag spectrometer program which must be running beforehand. In addition, it interfaces with 
a Labview program AutoTDS which records a platinum NMR thermometer trace as well as a 
melting curve thermometer reading and stores them under the directory named "Macintosh HD: 
current directory". Because a direct interface is difficult to program, the Applescript routine 
communicates with the Labview program through a file on the desktop named "filenumber". The 
information in this file is a single number which tell the Labview routine what number to add on to 
the name of the temperature data file. 



- set frequency, averages, file identifier for spectrometer 

set num to 1 as string 

tell application "setfiletoone" 

run 
end tell 
Mywait(30) 
tell application "setfiletoone" 

quit 
end tell 
Mywait(7) 

property storagefilename : "Macintosh HD:Desktop Folder:filenumber" 

set file_path to storagefilename 

set fileref to open for access file file_path 

set eof_position to get eof fileref 

set bignum to read fileref as text from 1 to eof_position 

set bignum to bignum div 2 

set bignum to bignum + 1 

set num to bignum as string 

close access fileref 

property storage_file_name2 : "Macintosh HD:Desktop Folder:iterations" 

set file_path to storage_file_name2 

set fileref to open for access file file_path 



338 



339 



set eof_position to get eof fileref 

set iter to read fileref as text fi:om 1 to eof_position 

set iter to iter as integer 

close access fileref 

repeat iter times 

tell application "autotds" 

run 
end tell 
Mywait(25) 
tell application "autotds" 

quit 
end tell 
Mywait(5) 

spectrometer( 105.6, 1056, 128, num) 
Mywait(lOO) 
tell application "autotds" 

run 
end tell 
Mywait(25) 
tell application "autotds" 

quit 
end tell 
Mywait(5) 
Mywait(360) 
set num to num + 1 

tell application "autotds" 

run 
end tell 
Mywait(25) 
tell application "autotds" 

quit 
end tell 
Mywait(5) 

spectrometer( 106.0, 1060, 128, num) 
Mywait(lOO) 
tell application "autotds" 

run 
end tell 
Mywait(25) 
tell application "autotds" 

quit 
end tell 
Mywait(5) 



340 



Mywait(360) 

set num to num + 1 

tell application "autotds" 

run 
end tell 
Mywait(25) 
tell application "autotds" 

quit 
end tell 
Mywait(5) 

spectrometer( 106.4, 1064, 128, num) 
Mywait(lOO) 
tell application "autotds" 

run 
end tell 
Mywait(25) 
tell application "autotds" 

quit 
end tell 
Mywait(5) 
Mywait(360) 
set num to num + 1 
tell application "autotds" 

nm 
end tell 
Mywait(25) 
tell application "autotds" 

quit 
end tell 
Mywait(5) 

spectrometer(106.8, 1068, 128, num) 
Mywait(lOO) 
tell application "autotds" 

run 
end tell 
Mywait(25) 
tell application "autotds" 

quit 
end tell 
Mywait(5) 
Mywait(360) 
set num to num + 1 
tell application "autotds" 

run 
end tell 



341 



Mywait(25) 

tell application "autotds" 

quit 
end tell 
Mywait(5) 

spectrometer( 107.2, 1072, 128, num) 
Mywait(lOO) 
tell application "autotds" 

run 
end tell 
Mywait(25) 
tell application "autotds" 

quit 
end tell 
Mywait(5) 
Mywait(360) 
set num to num + 1 
tell application "autotds" 

run 
end tell 
Mywait(25) 
tell application "autotds" 

quit 
end tell 
Mywait(5) 

spectrometer( 107.6, 1076, 128, num) 
Mywait(lOO) 
tell application "autotds" 

run 
end tell 
Mywait(25) 
tell application "autotds" 

quit 
end tell 
Mywait(5) 
Mywait(360) 
set num to num + 1 
tell application "autotds" 

run 
end tell 
Mywait(25) 
tell application "autotds" 

quit 
end tell 
Mywait(5) 



342 



spectrometer( 107.8, 1078, 128, num) 

Mywait(lOO) 

tell application "autotds" 
run 

end tell 

Mywait(25) 

tell application "autotds" 
quit 

end tell 

Mywait(5) 

Mywait(360) 

set num to num + 1 

tell application "autotds" 
run 

end tell 

Mywait(25) 

tell application "autotds" 
quit 

end tell 

Mywait(5) 

spectrometer( 108.0, 1080, 128, num) 

Mywait(lOO) 

tell application "autotds" 
run 

end tell 

Mywait(25) 

tell application "autotds" 

quit 
end tell 
Mywait(5) 
Mywait(360) 
set num to num + 1 
tell application "autotds" 

run 
end tell 
Mywait(25) 
tell application "autotds" 

quit 
end tell 
Mywait(5) 

spectrometer! 108.2, 1082, 128, num) 
Mywait(lOO) 
tell application "autotds" 

run 
end tell 



343 



Mywait(25) 

tell application "autotds" 

quit 
end tell 
Mywait(5) 
Mywait(360) 
set num to num + 1 
end repeat 

on spectrometer(frequency, nm, averages, niim) 
set freq to frequency as string 
teU application "MacNMR 5.5 PCI 1 1/06/97" 

activate 

Set NMR Parameter name "Obs Freq" value frequency 

Set NMR Parameter name "Scans ID" value averages 

Zero Memory 

Go 

repeat 

if (Check Acquisition) then exit repeat 

end repeat 

Baseline Correction 

Upload 

Baseline Correction 

set NMR to "Macintosh HDxurrentdata:" & nm & "M70p4s" & num 

save as {file NMR} 

set exp to NMR & ".EXP" 

Export as file exp type "TEXT" 
end tell 
end spectrometer 

on Mywait(t_in_seconds) 

set starttime to current date 

set deltatime to 

repeat until deltatime > tinseconds 

set deltatime to (current date) - starttime 

end repeat 

return deltatime 
end Mywait 



APPENDIX G 
ORIGIN SCRIPT 

This is an origin script that reads in typical data from the 2A pair breaking 
experiment that contains 9 different file names corresponding to 9 frequencies and blanks a 
portion of the received signal to zero. Then the script takes the Fourier transform of each 
received signal and saves the entire data set as an origin file. 

for(ii=59;ii<=100;ii-H-){ 

%V=$(ii); 

switch (mod((ii-l 3 ),9)) 

{ 
case 0: 

%A=1082M70p4s; 
case 1: 

%A=1056M70p4s; 
case 2: 

%A=1060M70p4s; 
case 3: 

%A=1064M70p4s; 
case 4: 

%A=1068M70p4s; 
case 5: 

%A=1072M70p4s; 
case 6: 

%A=1076M70p4s; 
case 7: 

%A=1078M70p4s; 
case 8: 

%A=1080M70p4s; 

}; 

window -n data; 

%F=A; 

%D=%A%V; 

%B = d:\02021tfscan\datafiles\%D.exp; 

open -w %B; 

worksheet -i 2 Mag; 



344 



345 



worksheet -n 1 X; 
worksheet -n 2 Y; 
worksheet -n 4 Time; 

for(iJ=laJ<=2048uJ++){ 
%F%D_Time[ij]=iJ* 1 E-4; 

}; 
for(ij=iaj<=i6ia^++){ 

%F%D_X[iJ]-0; 
%F%D_Y[iJ]=0; 

}; 

for(iJ=483aj<=2048aJ++){ 

%F%D_X[iJ]=0; 

%F%D_Y[iJ]-0; 

}; 
}; 

for(ii=59;ii<=100;ii++){ 
%V=$(ii); 

switch (mod((ii-13),9)) 
{ 

caseO: 

%A=1082M70p4s; 
case 1: 

%A=1056M70p4s; 
case 2: 

%A=1060M70p4s; 
case 3: 

%A=1064M70p4s; 
case 4: 

%A=1068M70p4s; 
case 5: 

%A=1072M70p4s; 
case 6: 

%A=1076M70p4s; 
case 7: 

%A=1078M70p4s; 
case 8: 

%A=1080M70p4s; 

}; 



346 



%F=A; 
%D=%A%V; 
window -a %F%D; 

ffl.resetO; 

ffl.forward^l; 

flft.forward.timeData$=%F%D_Time; 

fR.forward.tdelta=l E-4; 

ffl.forward.realData$=%F%D_X; 

ffi.forward.imagData$=%F%D_Y; 

window -t W fit %F%VFFT; 

ffl.output.samplingData$=%F%VFFT_Freq; 

ffl.output.powerData$=%F%VFFT_Power; 

ffl.output.realData$=%F%VFFT_Real; 

ffl.output.imagData$=%F%VFFT_Imag; 

ffl.output.ampData$=%F%VFFT_r; 

ffl.output.phaseData$=%F%VFFT_Phi; 

ffl.real=l; 

f!t.nonnalize=l; 

ffl.shifted=l; 

ffl.windowing=l; 

ffi.spectrum=l; 

ffl.unwrap=l; 

fit.forward(); 

worksheet -i 1 Amplitude; 
%F%VFFT_AmpUtude=%F%VFFT_Power; 
win -t plot c:\Origin50\FFT.otp %VFFT; 
win -i %F%D; 

layer -w %F%VFFT 1 1 2 2048; 
win -i %F%VFFT; 

}; 

type -h; 

window -s T; 

queue; 

for(ii=l;ii<=25;ii++){ 

type-b; 

}; 

save d:\02021tfscan\graphs\02021tblank59- 1 00; 
type -b The Script is Finished.; 



APPENDIX H 
MATLAB FITTING PROGRAMS 



There are two matlab programs in this appendix. Details of these programs are 
given in Chapter 3 of this thesis. The first program, entitled "mfit", performs an exact 
diagonalization fit to data which is described by a polynomial or other fimction. The 
program calls a subroutine entitled "magladB" which was written by S.E. Nagler at Oak 
Ridge National Laboratory. For information regarding this subroutine, please contact him 
using the email address, naglerse@oml.gov. The second program, entitled "fi-ee", creates 
magnetization curves by calculating the fi-ee energy of a ladder by mapping the ladder 
Hamiltonian onto the exactly solvable XXZ chain. 



fimction mfit(g,Jperpstart,Jperpend,Jparstart,Jparend,T) 
B=(0:.01:30); 
A=B*(.671744*g); 
numb==l; 

mcalc=(0:.01:40); 
mbest=lE16; 
Jperpbest=0; 
Jparbest==0; 

fid=fopen(' 1 21ad 1 75 .doc', V); 
disk=(l:l:3); 
format compact 

for G=Jperpstart:0.05:Jperpend 
for H=Jparstart:0.05:Jparend 
mchi=0; 

mag=magladB(6,G,H,0,T,A); 
forK=701:l:1701 
mcalc(K)=(2 1 366- 1 2807*(B(K))+3045 . 1 *(B(K))^2- 
365. 1 7*(B(K))'^3+24. 1 84*(B(K))M- 

0.83483*(B(K)r5+.011666*(B(K)r6)/12012; 
mchi=mchi+(mag(K)-mcalc(K))^2; 
%mcalc(K) 
%mag(K) 
end 
disk=[G H mchi]; 



347 



348 



count=fprintf(l,'%g %g %g \n',disk); 
count=^rintfi:ficl,'%g %g %g \n',disk); 
%Jperpbest,Jparbest,mbest 
ifmchi<mbest 
mbest=mchi; 
Jperpbest=G; 
Jparbest=H; 

disk=[Jperpbest Jparbest mbest]; 
count=^rintf(l,'%g %g %g \n',disk); 
numb=numb+l; 

%plot(A,mag,'B-',B,mcalc,'R-'); 
%pause(7); 
end 
end 
end 

disk=[Jperpbest Jparbest mbest]; 
count=irintf(fid,'%g %g %g \n',disk); 

status=fclose('all') 
return 



function [F]=free(T) 

Jl=13.27; 

J2=3.80; 

form =1:1:601 

H=(m-l)/20; 

m 

F(m)=grenob(2. 1 3,J1 ,J2,T,H); 

fit(m)=0; 
end 

LK0:.05:30); 

E=-difi(F); 

const=-E(l); 

E=E+const; 

scale=6006/E(600); 

E(601)=E(600); 

E=E*scale; 

chi=0; 

forp= 171:1:251 
H=(p-l)/20; 



349 



fit(p)=78081 . 12-44491 . 63*(H+0.025)+9924.65*(H+0.025)^2- 
1104.934*(H+0.025)^3 

+65.00669*(H+0.025)M-1.882572*(H+0.025)^5+.02016716*(H+0.025)%; 
chi=chi+(fit(p)-E(p))^2; 
end 
chi 
plot(L,E,'B-',L,fit,'R-') 

fid=fopen('free075.dat','w'); 
for p= 1:1:601 

H=(p-l)/2+0.025; 

disk=[H E(p)]; 

count=^rintf(fid,'%g %g \n',disk); 
end 

status=fclose('all') 
return 

function [G]=grenob{g,Jperp,Jpar,T,H) 
HefiNJperp+(Jpar/2)-g*0.67 1 744*H; 

for 1=1:1:201 

n(i)=3; 

s(i)=0.25*sech(3.1415927*(i-101)/20); 

k(i)=2; 
end 

G=0; 

fori= 1:1:15 

y=log(l+n); 

k=exp(convol(s,y)); 

u=2*k*cosh(l .5*HeflB'T)+k.*k; 

z=Iog(l+u); 

n=exp(-5.196152*Jpar*s/T+convol(s,z)); 
end 

G=-Jpar/2-T*log(k(101)); 
%^rintf(l,'%g\n',G); 
return 

function [C]=convoI(A,B) 

C=conv(A,B); 

C=C(101:1:301); 

C=C/10; 
return 



APPENDIX I 
DATA SET PARAMETERS FOR LANDAU LIMIT EXPERIMENT 



Table I.l: Zero sound data taken at the pressure of 1 bar in the above order using the 
frequencies of 10, 16, 19, 20, 21, 22, 23, 63, 64, 65, 66, 66.6, 107, and 108 MHz. The 
data set ZSIO was used for the subtraction of the electrical crosstalk. 



Data Set Name 


Tstart (mK) 


Tend (mK) 


Tave (mK) 


ZSl 


2.293 


2.309 


2.301 


ZS2 


2.499 


2.540 


2.520 


ZS3 


1.706 


1.729 


1.718 


ZS4 


1.116 


1.102 


1.109 


ZS5 


1.183 


1.215 


1.199 


ZS6 


1.436 


1.475 


1.455 


ZS7 


2.010 


2.050 


2.030 


ZS8 


2.738 


2.784 


2.761 


ZS9 


2.940 


2.854 


2.898 


ZSIO 


3.484 


3.568 


3.526 



350 



351 



Table 1.2: First sound data taken at the pressure of 1 bar in the above order using the 
frequencies of 10, 16, 19, 20, 21, 22, 23, 63, 64, 65, 66, 66.6, 107, and 108 MHz. The 
data set HN2 was used for the subtraction of the electrical crosstalk. The data set HN13 
was accidentally skipped. 



Data Set Name 


Tstart (mK) 


Tend (mK) 


Tave (mK) 


HNl 


48.33698 


48.39253 


48.997 


HN2 


28.8822 


29.01252 


29.683 


HN3 


57.59679 


57.66121 


58.129 


HN4 


70.40064 


70.50365 


70.461 


HNS 


82.29028 


82.42592 


83.278 


HN6 


100.17435 


100.07867 


100.550 


HN7 


115.96167 


116.02804 


116.168 


HN8 


145.04688 


145.11459 


144.791 


HN9 


202.85107 


202.76667 


202.411 


HNIO 


261.29311 


261.06694 


262.480 


HNll 


380.493 


382.124 


381.308 


HN12 


465.289 


465.343 


465.328 


HN14 


604.907 


604.909 


604.922 


HN15 


691.105 


691.135 


691.132 


HN16 


747.941 


747.965 


747.913 


HN17 


720.504 


720.491 


720.495 


HN18 


650.833 


650.849 


650.862 


HN19 


537.899 


537.874 


537.941 


HN20 


804.782 


804.801 


804.741 


HN21 


266.305 


261.893 


333.821 


HN22 


495.754 


495.743 


495.748 



352 



Table 1.3: Zero sound data taken at the pressure of 5 bars in the above order at the 
frequencies of 8, 10, 16, 20, 23, 28, 63, 64, 65, 66.6, 68, 106, 107, 108 and 109 MHz. 
The data set ZS6 was measured at approximately 6 mK, which is outside the calibration 
range of the melting curve thermometer, and was used for the subtraction of the electrical 
crosstalk. 



Data Set Name 


Tstert (mK) 


Tend (mK) 


Tave (mK) 


ZSl 


2.287 


2.300 


2.293 


ZS2 


2.363 


2.379 


2.371 


ZS3 


1.954 


1.966 


1.960 


ZS4 


1.734 


1.746 


1.740 


ZS5 


1.546 


1.557 


1.552 


ZS6 


- 


- 


- 


ZS7 


4.425 


4.473 


4.449 


ZS8 


3.624 


3.649 


3.636 


ZS9 


3.199 


3.235 


3.217 


ZSIO 


2.941 


2.968 


2.955 


ZSll 


2.708 


2.755 


2.731 


ZS12 


2.540 


2.561 


2.550 


ZS13 


2.198 


2.219 


2.209 


ZS14 


1.989 


2.011 


2.000 


ZS15 


1.792 


1.822 


1.807 


ZS16 


2.715 


2.753 


2.734 


ZS17 


2.728 


2.814 


2.771 



353 



Table 1.4: First sound data taken at the pressure of 5 bars in the above order at the 
frequencies of 8, 10, 16, 20, 23, 28, 63, 64, 65, 66.6, 68, 106, 107, 108 and 109 MHz. 
The data set HF3 was used for the subtraction of the electrical crosstalk. 



Data Set Name 


Tstart (mK) 


Tend (mK) 


Tave (mK) 


HFl 


27.999 


28.096 


28.047 


HF2 


23.657 


23.669 


23.663 


HF3 


19.355 


19.364 


19.359 


HF4 


40.364 


40.425 


40.395 


HF5 


49.554 


49.580 


49.567 


HF6 


59.164 


59.186 


59.175 


HF7 


69.332 


69.377 


69.355 


HF8 


80.556 


80.673 


80.614 


HF9 


93.036 


93.049 


93.043 


HFIO 


99.733 


99.751 


99.742 


HFll 


140.307 


140.218 


140.263 


HF12 


200.704 


201.337 


201.020 


HF13 


241.419 


243.218 


242.318 


HF14 


317.919 


312.621 


315.270 


HF15 


336.951 


334.420 


335.686 


HF16 


391.077 


391.046 


391.062 


HF17 


449.824 


450.163 


449.994 


HF18 


503.561 


531.254 


517.408 


HF19 


346.912 


347.102 


347.007 


HF20 


500.438 


500.521 


500.480 


HF21 


550.469 


550.375 


550.422 


HF22 


599.977 


599.971 


599.974 


HF23 


649.971 


649.978 


649.974 


HF24 


700.166 


700.140 


700.153 


HF25 


759.831 


759.861 


759.846 


HF26 


775.238 


775.191 


775.215 


HF27 


795.671 


795.658 


795.664 



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BIOGRAPHICAL SKETCH 

Brian Watson was bom near in Blue Island, Illinois, as the second youngest of five 
siblings. At age 12, he had his own workshop for building science projects and in 
elementary school won the science fair more often than not. Brian graduated fi-om 
Eisenhower High School in Blue Island, Illinois in 1986. That same year he began college 
at the University of Illinois at Champaign-Urbana and graduated in 1990 with a bachelor's 
degree in electrical engineering. In 1994, after working for the Navy and Air Force for 
almost four years as an electronics engineer, he applied and was accepted to graduate 
school in physics at Southern Illinois University, Carbondale, Illinois. The topic of his 
master's thesis was the magnetic properties of rare earth metals. After receiving his 
master's degree in physics in 1995, and because Southern Illinois did not offer a doctorate 
in physics, he applied to the University of Florida physics department. In the fall of 1996, 
he began school at the University of Florida, and in the spring of 1997, joined Professor 
Meisel's research laboratory. This dissertation is a culmination of his research efforts over 
the past three years. 



362 



I certify that I have read this study and tliat, in my opinion, it confirms to 
acceptable standards of scholarly presentation and is fully adequate, in scope and quality, 
as a dissertation for the degree of Doctor of Philosophy. 




Mark W. Meisel, Chairman 
Professor of Physics 



I certify that I have read this study and that, in my opinion, it confirms to 
acceptable standards of scholarly presentation and is fully adequate, in scope and quality, 
as a dissertation for the degree of Doctor of Philosophy. 



rthur F. Hebard C_^ 



Arthur 

Professor of Physics 



I certify that I have read this study and that, in my opinion, it confirms to 
acceptable standards of scholarly presentation and is fiilly adequate, in scope and quality, 
as a dissertation for the degree of Doctor of Philosophy. 

1 U^ ■ 

Kevin Inger^t 

Associate Professor of Physics 



I certify that I have read this study and that, in my opinion, it confirms to 
acceptable standards of scholarly presentation and is fiilly adequate, in scope and quality, 
as a dissertation for the degree of Doctor of Philosophy. 





yYasumasa Takano 
Professor of Physics 



I certify that I have read this study and that, in my opinion, it confirms to 
acceptable standards of scholarly presentation and is flilly adequate, in scope and quality, 
as a dissertation for the degree of Doctor of Philosophy. 




Daniel R. Talham 
Professor of Chemistry 



I certify that I have read this study and that, in my opinion, it conilrms to 
acceptable standards of scholarly presentation and is fully adequate, in scope and quality, 
as a dissertation for the degree of Doctor of Philosophy. 




Mark W. Meisel, Chairman 
Professor of Physics 



I certify that I have read this study and that, in my opinion, it confirms to 
acceptable standards of scholarly presentation and is fully adequate, in scope and quality, 
as a dissertation for the degree of Doctor of Philosophy. 




Arthur F. Hebard 
Professor of Physics 



I certify that I have read this study and that, in my opinion, it confirms to 
acceptable standards of scholarly presentation and is fijlly adequate, in scope and quality, 
as a dissertation for the degree of Doctor of Philosophy. 




Kevin Ingersent 

Associate Professor of Physics 



I certify that I have read this study and that, in my opinion, it confirms to 
acceptable standards of scholarly presentation and is fully adequate, in scope and quality, 
as a dissertation for the degree of Doctor of Philosophy. 




^A^umasa Takano 
Professor of Physics 



I certify that I have read this study and that, in my opinion, it confirms to 
acceptable standards of scholarly presentation and is fully adequate, in scope and quality, 
as a dissertation for the degree of Doctor of Philosophy. 




Daniel R. Talham 
Professor of Chemistry 



i~0 



UNIVERSITY OF FLORIDA 



3 1262 08555 2007